Abstract group $C_2^3.C_2^5$

• Label: 256.49985
• Order: \( 2^{8} \)
• Exponent: \( 2^{2} \)
• Nilpotent: yes
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{5} \)
• $\card{Z(G)}$: \( 2^{3} \)
• $\card{\mathrm{Aut}(G)}$: \( 2^{15} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{10} \)
• Perm deg.: $40$
• Trans deg.: $128$
• Rank: $5$

Group information

Description:	$C_2^3.C_2^5$
Order:	\(256\)\(\medspace = 2^{8} \)
Exponent:	\(4\)\(\medspace = 2^{2} \)
Automorphism group:	Group of order \(32768\)\(\medspace = 2^{15} \) (generators)
Outer automorphisms:	$C_2^{10}$, of order \(1024\)\(\medspace = 2^{10} \)
Composition factors:	$C_2$ x 8
Nilpotency class:	$2$
Derived length:	$2$

This group is nonabelian, a $p$-group (hence nilpotent, solvable, supersolvable, monomial, elementary, and hyperelementary), and metabelian.

Group statistics

Order	1	2	4
Elements	1	39	216	256
Conjugacy classes	1	11	34	46
Divisions	1	11	28	40
Autjugacy classes	1	11	27	39

Dimension	1	4	8
Irr. complex chars.	32	14	0	46
Irr. rational chars.	32	2	6	40

Minimal Presentations

Permutation degree:	$40$
Transitive degree:	$128$
Rank:	$5$
Inequivalent generating 5-tuples:	$9999360$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	none	none	none
Arbitrary	12	20	20

Constructions

Presentation:	${\langle a, b, c, d, e \mid b^{2}=c^{4}=d^{4}=e^{4}=[a,c]=[b,e]=[c,d]=[d,e]= \!\cdots\! \rangle}$
Permutation group:	Degree $40$ $\langle(1,2)(3,12)(4,9)(5,10)(6,13)(7,11)(8,15)(14,16)(17,18,21,19)(20,23,24,22) \!\cdots\! \rangle$
Direct product:	not isomorphic to a non-trivial direct product
Semidirect product:	$(C_4^2.D_4)$ $\,\rtimes\,$ $C_2$	$(C_2^3.C_2^4)$ $\,\rtimes\,$ $C_2$	$(C_2^3.C_2^4)$ $\,\rtimes\,$ $C_2$	$(C_2^3.C_2^4)$ $\,\rtimes\,$ $C_2$	all 29
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$C_2^4$ . $C_2^4$ (4)	$C_2^3$ . $C_2^5$	$(C_2^3.Q_8)$ . $C_2^2$ (8)	$(C_2^3.Q_8)$ . $C_2^2$ (5)	all 43

Elements of the group are displayed as words in the generators from the presentation given above.

Homology

Abelianization:	$C_{2}^{5} $
Schur multiplier:	$C_{2}^{7}$
Commutator length:	$1$

Subgroups

There are 1559 subgroups in 725 conjugacy classes, 391 normal (389 characteristic).

Characteristic subgroups are shown in this color. Normal (but not characteristic) subgroups are shown in this color.

Special subgroups

Center:	$Z \simeq$ $C_2^3$	$G/Z \simeq$ $C_2^5$
Commutator:	$G' \simeq$ $C_2^3$	$G/G' \simeq$ $C_2^5$
Frattini:	$\Phi \simeq$ $C_2^3$	$G/\Phi \simeq$ $C_2^5$
Fitting:	$\operatorname{Fit} \simeq$ $C_2^3.C_2^5$	$G/\operatorname{Fit} \simeq$ $C_1$
Radical:	$R \simeq$ $C_2^3.C_2^5$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_2^3$	$G/\operatorname{soc} \simeq$ $C_2^5$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2^3.C_2^5$

Hi

Subgroup diagram and profile

Classes of subgroups up to conjugation

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Classes of subgroups up to automorphism

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Normal subgroups

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Normal subgroups up to automorphism

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Classes of subgroups up to conjugation

Order 256: $C_2^3.C_2^5$

Order 128: $C_2^3.C_2^4$ x 2, $C_2^3.C_2^4$, $C_2^3.C_2^4$, $C_2^3.C_2^4$, $C_2^3.C_2^4$, $C_2^3.C_2^4$, $C_2^3.C_2^4$, $C_2^3.C_2^4$, $C_2^3.C_2^4$, $C_2^3.C_2^4$, $C_2^3.C_2^4$, $C_2^3.C_2^4$, $C_2^3.C_2^4$, $C_2^3.C_2^4$, $C_2^3.C_2^4$, $C_2^3.C_2^4$, $C_2^3.C_2^4$, $C_2^3.C_2^4$, $C_2^3.C_2^4$, $C_2^3.C_2^4$, $C_4^2.D_4$, $C_2^3.C_2^4$, $C_2^3.C_2^4$, $C_2^3.C_2^4$, $C_2^3.C_2^4$, $C_2^3.C_2^4$, $C_2^3.C_2^4$, $C_2^3.C_2^4$, $C_2^3.C_2^4$, $C_2^3.C_2^4$

Order 64: $C_2^3.C_2^3$ x 23, $C_2^3.C_2^3$ x 17, $C_2^3.D_4$ x 15, $C_2^3.D_4$ x 14, $C_2^3.C_2^3$ x 11, $C_2^3.C_2^3$ x 8, $C_2^3.Q_8$ x 8, $C_2^3.C_2^3$ x 8, $C_2^3.D_4$ x 7, $C_2^3.D_4$ x 7, $C_2^3.C_2^3$ x 6, $C_2^3.Q_8$ x 5, $C_2^3.D_4$ x 4, $C_2^3:D_4$ x 3, $C_2^3.D_4$ x 3, $C_2^2:C_4^2$ x 3, $C_4^2:C_4$ x 2, $C_4^2:C_4$ x 2, $C_4:C_4^2$ x 2, $C_4^2.C_2^2$ x 2, $C_4^2:C_4$, $C_4^2:C_2^2$, $C_2^3.D_4$, $C_2^3:D_4$, $C_4^2:C_2^2$

Order 32: $C_2.C_4^2$ x 52, $C_2^3:C_4$ x 44, $C_2^2.D_4$ x 38, $C_2\times C_4^2$ x 11, $C_2^2\times D_4$ x 6, $C_2^3\times C_4$ x 4, $C_4.Q_8$ x 4, $C_4^2:C_2$ x 2, $C_2^2.D_4$ x 2, $C_4:D_4$ x 2, $C_4\times D_4$ x 2

Order 16: $C_2^2:C_4$ x 45, $C_4:C_4$ x 43, $C_2^2\times C_4$ x 40, $C_2\times D_4$ x 19, $C_4^2$ x 12, $C_2^4$ x 4

Order 8: $C_2\times C_4$ x 90, $C_2^3$ x 29, $D_4$ x 14

Order 4: $C_2^2$ x 35, $C_4$ x 28

Order 2: $C_2$ x 11

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 256: $C_2^3.C_2^5$

Order 128: $C_2^3.C_2^4$ x 2, $C_2^3.C_2^4$, $C_2^3.C_2^4$, $C_2^3.C_2^4$, $C_2^3.C_2^4$, $C_2^3.C_2^4$, $C_2^3.C_2^4$, $C_2^3.C_2^4$, $C_2^3.C_2^4$, $C_2^3.C_2^4$, $C_2^3.C_2^4$, $C_2^3.C_2^4$, $C_2^3.C_2^4$, $C_2^3.C_2^4$, $C_2^3.C_2^4$, $C_2^3.C_2^4$, $C_2^3.C_2^4$, $C_2^3.C_2^4$, $C_2^3.C_2^4$, $C_2^3.C_2^4$, $C_4^2.D_4$, $C_2^3.C_2^4$, $C_2^3.C_2^4$, $C_2^3.C_2^4$, $C_2^3.C_2^4$, $C_2^3.C_2^4$, $C_2^3.C_2^4$, $C_2^3.C_2^4$, $C_2^3.C_2^4$, $C_2^3.C_2^4$

Order 64: $C_2^3.C_2^3$ x 23, $C_2^3.C_2^3$ x 17, $C_2^3.D_4$ x 15, $C_2^3.D_4$ x 14, $C_2^3.C_2^3$ x 11, $C_2^3.C_2^3$ x 8, $C_2^3.Q_8$ x 8, $C_2^3.C_2^3$ x 8, $C_2^3.D_4$ x 7, $C_2^3.D_4$ x 7, $C_2^3.C_2^3$ x 6, $C_2^3.Q_8$ x 5, $C_2^3.D_4$ x 4, $C_2^3:D_4$ x 3, $C_2^3.D_4$ x 3, $C_2^2:C_4^2$ x 3, $C_4^2:C_4$ x 2, $C_4^2:C_4$ x 2, $C_4:C_4^2$ x 2, $C_4^2.C_2^2$ x 2, $C_4^2:C_4$, $C_4^2:C_2^2$, $C_2^3.D_4$, $C_2^3:D_4$, $C_4^2:C_2^2$

Order 32: $C_2.C_4^2$ x 52, $C_2^3:C_4$ x 44, $C_2^2.D_4$ x 38, $C_2\times C_4^2$ x 11, $C_2^2\times D_4$ x 6, $C_2^3\times C_4$ x 4, $C_4.Q_8$ x 2, $C_4^2:C_2$, $C_2^2.D_4$, $C_4:D_4$, $C_4\times D_4$

Order 16: $C_2^2:C_4$ x 44, $C_2^2\times C_4$ x 39, $C_4:C_4$ x 38, $C_2\times D_4$ x 18, $C_4^2$ x 11, $C_2^4$ x 4

Order 8: $C_2\times C_4$ x 85, $C_2^3$ x 29, $D_4$ x 6

Order 4: $C_2^2$ x 35, $C_4$ x 27

Order 2: $C_2$ x 11

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 256: $C_2^3.C_2^5$ ($C_1$)

Order 128: $C_2^3.C_2^4$ ($C_2$) x 2, $C_2^3.C_2^4$ ($C_2$), $C_2^3.C_2^4$ ($C_2$), $C_2^3.C_2^4$ ($C_2$), $C_2^3.C_2^4$ ($C_2$), $C_2^3.C_2^4$ ($C_2$), $C_2^3.C_2^4$ ($C_2$), $C_2^3.C_2^4$ ($C_2$), $C_2^3.C_2^4$ ($C_2$), $C_2^3.C_2^4$ ($C_2$), $C_2^3.C_2^4$ ($C_2$), $C_2^3.C_2^4$ ($C_2$), $C_2^3.C_2^4$ ($C_2$), $C_2^3.C_2^4$ ($C_2$), $C_2^3.C_2^4$ ($C_2$), $C_2^3.C_2^4$ ($C_2$), $C_2^3.C_2^4$ ($C_2$), $C_2^3.C_2^4$ ($C_2$), $C_2^3.C_2^4$ ($C_2$), $C_2^3.C_2^4$ ($C_2$), $C_4^2.D_4$ ($C_2$), $C_2^3.C_2^4$ ($C_2$), $C_2^3.C_2^4$ ($C_2$), $C_2^3.C_2^4$ ($C_2$), $C_2^3.C_2^4$ ($C_2$), $C_2^3.C_2^4$ ($C_2$), $C_2^3.C_2^4$ ($C_2$), $C_2^3.C_2^4$ ($C_2$), $C_2^3.C_2^4$ ($C_2$), $C_2^3.C_2^4$ ($C_2$)

Order 64: $C_2^3.C_2^3$ ($C_2^2$) x 23, $C_2^3.C_2^3$ ($C_2^2$) x 17, $C_2^3.D_4$ ($C_2^2$) x 15, $C_2^3.D_4$ ($C_2^2$) x 14, $C_2^3.C_2^3$ ($C_2^2$) x 11, $C_2^3.C_2^3$ ($C_2^2$) x 8, $C_2^3.Q_8$ ($C_2^2$) x 8, $C_2^3.C_2^3$ ($C_2^2$) x 8, $C_2^3.D_4$ ($C_2^2$) x 7, $C_2^3.D_4$ ($C_2^2$) x 7, $C_2^3.C_2^3$ ($C_2^2$) x 6, $C_2^3.Q_8$ ($C_2^2$) x 5, $C_2^3.D_4$ ($C_2^2$) x 4, $C_2^3:D_4$ ($C_2^2$) x 3, $C_2^3.D_4$ ($C_2^2$) x 3, $C_2^2:C_4^2$ ($C_2^2$) x 3, $C_4^2:C_4$ ($C_2^2$) x 2, $C_4^2:C_4$ ($C_2^2$) x 2, $C_4:C_4^2$ ($C_2^2$) x 2, $C_4^2.C_2^2$ ($C_2^2$) x 2, $C_4^2:C_4$ ($C_2^2$), $C_4^2:C_2^2$ ($C_2^2$), $C_2^3.D_4$ ($C_2^2$), $C_2^3:D_4$ ($C_2^2$), $C_4^2:C_2^2$ ($C_2^2$)

Order 32: $C_2.C_4^2$ ($C_2^3$) x 52, $C_2^3:C_4$ ($C_2^3$) x 44, $C_2^2.D_4$ ($C_2^3$) x 38, $C_2\times C_4^2$ ($C_2^3$) x 11, $C_2^2\times D_4$ ($C_2^3$) x 6, $C_2^3\times C_4$ ($C_2^3$) x 4

Order 16: $C_2^2\times C_4$ ($C_2^4$) x 27, $C_2^4$ ($C_2^4$) x 4

Order 8: $C_2\times C_4$ ($D_4:C_2^2$) x 2, $C_2^3$ ($C_2^5$)

Order 4: $C_2^2$ ($C_4.C_2^4$) x 6, $C_2^2$ ($D_4:C_2^3$)

Order 2: $C_2$ ($C_4^2.C_2^3$) x 3, $C_2$ ($C_4^2.C_2^3$), $C_2$ ($C_4^2.C_2^3$), $C_2$ ($C_4^2.C_2^3$), $C_2$ ($C_4^2.C_2^3$)

Order 1: $C_1$ ($C_2^3.C_2^5$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 256: $C_2^3.C_2^5$ ($C_1$)

Order 128: $C_2^3.C_2^4$ ($C_2$) x 2, $C_2^3.C_2^4$ ($C_2$), $C_2^3.C_2^4$ ($C_2$), $C_2^3.C_2^4$ ($C_2$), $C_2^3.C_2^4$ ($C_2$), $C_2^3.C_2^4$ ($C_2$), $C_2^3.C_2^4$ ($C_2$), $C_2^3.C_2^4$ ($C_2$), $C_2^3.C_2^4$ ($C_2$), $C_2^3.C_2^4$ ($C_2$), $C_2^3.C_2^4$ ($C_2$), $C_2^3.C_2^4$ ($C_2$), $C_2^3.C_2^4$ ($C_2$), $C_2^3.C_2^4$ ($C_2$), $C_2^3.C_2^4$ ($C_2$), $C_2^3.C_2^4$ ($C_2$), $C_2^3.C_2^4$ ($C_2$), $C_2^3.C_2^4$ ($C_2$), $C_2^3.C_2^4$ ($C_2$), $C_2^3.C_2^4$ ($C_2$), $C_4^2.D_4$ ($C_2$), $C_2^3.C_2^4$ ($C_2$), $C_2^3.C_2^4$ ($C_2$), $C_2^3.C_2^4$ ($C_2$), $C_2^3.C_2^4$ ($C_2$), $C_2^3.C_2^4$ ($C_2$), $C_2^3.C_2^4$ ($C_2$), $C_2^3.C_2^4$ ($C_2$), $C_2^3.C_2^4$ ($C_2$), $C_2^3.C_2^4$ ($C_2$)

Order 64: $C_2^3.C_2^3$ ($C_2^2$) x 23, $C_2^3.C_2^3$ ($C_2^2$) x 17, $C_2^3.D_4$ ($C_2^2$) x 15, $C_2^3.D_4$ ($C_2^2$) x 14, $C_2^3.C_2^3$ ($C_2^2$) x 11, $C_2^3.C_2^3$ ($C_2^2$) x 8, $C_2^3.Q_8$ ($C_2^2$) x 8, $C_2^3.C_2^3$ ($C_2^2$) x 8, $C_2^3.D_4$ ($C_2^2$) x 7, $C_2^3.D_4$ ($C_2^2$) x 7, $C_2^3.C_2^3$ ($C_2^2$) x 6, $C_2^3.Q_8$ ($C_2^2$) x 5, $C_2^3.D_4$ ($C_2^2$) x 4, $C_2^3:D_4$ ($C_2^2$) x 3, $C_2^3.D_4$ ($C_2^2$) x 3, $C_2^2:C_4^2$ ($C_2^2$) x 3, $C_4^2:C_4$ ($C_2^2$) x 2, $C_4^2:C_4$ ($C_2^2$) x 2, $C_4:C_4^2$ ($C_2^2$) x 2, $C_4^2.C_2^2$ ($C_2^2$) x 2, $C_4^2:C_4$ ($C_2^2$), $C_4^2:C_2^2$ ($C_2^2$), $C_2^3.D_4$ ($C_2^2$), $C_2^3:D_4$ ($C_2^2$), $C_4^2:C_2^2$ ($C_2^2$)

Order 32: $C_2.C_4^2$ ($C_2^3$) x 52, $C_2^3:C_4$ ($C_2^3$) x 44, $C_2^2.D_4$ ($C_2^3$) x 38, $C_2\times C_4^2$ ($C_2^3$) x 11, $C_2^2\times D_4$ ($C_2^3$) x 6, $C_2^3\times C_4$ ($C_2^3$) x 4

Order 16: $C_2^2\times C_4$ ($C_2^4$) x 27, $C_2^4$ ($C_2^4$) x 4

Order 8: $C_2^3$ ($C_2^5$), $C_2\times C_4$ ($D_4:C_2^2$)

Order 4: $C_2^2$ ($C_4.C_2^4$) x 6, $C_2^2$ ($D_4:C_2^3$)

Order 2: $C_2$ ($C_4^2.C_2^3$) x 3, $C_2$ ($C_4^2.C_2^3$), $C_2$ ($C_4^2.C_2^3$), $C_2$ ($C_4^2.C_2^3$), $C_2$ ($C_4^2.C_2^3$)

Order 1: $C_1$ ($C_2^3.C_2^5$)

Series

Derived series

$C_2^3.C_2^5$

$\rhd$

$C_2^3$

$\rhd$

$C_1$

Chief series

$C_2^3.C_2^5$

$\rhd$

$C_2^3.C_2^4$

$\rhd$

$C_4^2:C_4$

$\rhd$

$C_2\times C_4^2$

$\rhd$

$C_2^2\times C_4$

$\rhd$

$C_2^3$

$\rhd$

$C_2^2$

$\rhd$

$C_2$

$\rhd$

$C_1$

Lower central series

$C_2^3.C_2^5$

$\rhd$

$C_2^3$

$\rhd$

$C_1$

Upper central series

$C_1$

$\lhd$

$C_2^3$

$\lhd$

$C_2^3.C_2^5$

Character theory

Complex character table

See the $46 \times 46$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $40 \times 40$ rational character table.