Abstract group 256.53382: $C_2^3.C_2^5$

• Label: 256.53382
• Order: \( 2^{8} \)
• Exponent: \( 2^{2} \)
• Nilpotent: yes
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{5} \)
• $\card{Z(G)}$: \( 2^{3} \)
• $\card{\Aut(G)}$: \( 2^{15} \cdot 3 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{10} \cdot 3 \)
• Perm deg.: $18$
• Trans deg.: $32$
• Rank: $5$

Group information

Description:	$C_2^3.C_2^5$
Order:	\(256\)\(\medspace = 2^{8} \)
Exponent:	\(4\)\(\medspace = 2^{2} \)
Automorphism group:	$C_2^4.C_2^4.C_6.C_2^5.C_2$, of order \(98304\)\(\medspace = 2^{15} \cdot 3 \)
Composition factors:	$C_2$ x 8
Nilpotency class:	$3$
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	79	176	256
Conjugacy classes	1	25	38	64
Divisions	1	25	34	60
Autjugacy classes	1	6	5	12

Dimension	1	2	4	8
Irr. complex chars.	32	24	8	0	64
Irr. rational chars.	32	24	0	4	60

Minimal presentations

Permutation degree:	$18$
Transitive degree:	$32$
Rank:	$5$
Inequivalent generating 5-tuples:	$3333120$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	none	none	none
Arbitrary	not computed	not computed	not computed

Constructions

Presentation:	${\langle a, b, c, d, e, f \mid a^{2}=b^{2}=c^{2}=d^{2}=e^{4}=f^{4}=[a,b]= \!\cdots\! \rangle}$
Permutation group:	Degree $18$ $\langle(6,12)(13,16), (1,2)(3,6)(4,7)(5,8)(9,13)(10,12)(11,14)(15,16), (1,3)(2,6) \!\cdots\! \rangle$
Transitive group:	32T3926	32T4706	32T4709		more information
Direct product:	$C_2$ $\, \times\, $ $(C_2^3.C_2^4)$
Semidirect product:	$(C_2^2\times Q_8)$ $\,\rtimes\,$ $D_4$	$(C_2^2\times D_4)$ $\,\rtimes\,$ $D_4$	$(D_4:C_2^2)$ $\,\rtimes\,$ $D_4$	$(C_2^4:D_4)$ $\,\rtimes\,$ $C_2$	all 27
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$C_2^5$ . $C_2^3$	$C_2^4$ . $C_2^4$ (2)	$C_2^3$ . $C_2^5$	$C_2^4$ . $(C_2\times D_4)$	all 23
Aut. group:	$\Aut(C_2\times \OD_{32})$	$\Aut(\OD_{32}:C_6)$

Elements of the group are displayed as permutations of degree 18.

Homology

Abelianization:	$C_{2}^{5} $
Schur multiplier:	$C_{2}^{10}$
Commutator length:	$1$

Subgroups

There are 5199 subgroups in 2543 conjugacy classes, 559 normal (12 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_2\times C_4$	$G/Z \simeq$ $C_2^2\wr C_2$
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:	$\operatorname{soc} \simeq$ $C_2^2$	$G/\operatorname{soc} \simeq$ $C_2^3:D_4$
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 16, $C_2^4.D_4$ x 4, $C_2^4:D_4$ x 4, $C_4^2:C_2^3$ x 3, $C_2^3.C_2^4$ x 3, $C_4.C_2^5$

Order 64: $C_4^2:C_2^2$ x 36, $C_2^3.D_4$ x 32, $C_2\wr C_2^2$ x 32, $C_2^3.C_2^3$ x 24, $C_4.C_2^4$ x 20, $C_2^4:C_4$ x 12, $D_4:C_2^3$ x 9, $D_4:C_2^3$ x 7, $C_2^3.D_4$ x 6, $C_2^3:D_4$ x 6, $C_4^2:C_2^2$ x 6, $C_4.C_2^4$ x 3, $C_2^3:Q_8$ x 3, $C_2^3:D_4$ x 3, $C_4^2:C_2^2$ x 3, $C_2^4\times C_4$

Order 32: $D_4:C_2^2$ x 172, $D_4:C_2^2$ x 88, $C_2^3:C_4$ x 48, $C_4.C_2^3$ x 48, $C_4\times D_4$ x 48, $C_2^2.D_4$ x 36, $C_2^2\wr C_2$ x 36, $C_2^2\times D_4$ x 33, $C_2^3\times C_4$ x 28, $C_2^2:Q_8$ x 24, $C_4:D_4$ x 24, $C_4^2:C_2$ x 18, $C_2^3:C_4$ x 18, $C_2^2\times Q_8$ x 9, $C_2^2.D_4$ x 9, $C_2\times C_4^2$ x 3, $C_2^5$

Order 16: $D_4:C_2$ x 320, $C_2\times D_4$ x 276, $C_2^2\times C_4$ x 166, $C_2\times Q_8$ x 76, $C_2^2:C_4$ x 66, $C_4:C_4$ x 36, $C_2^4$ x 23, $C_4^2$ x 12

Order 8: $C_2\times C_4$ x 198, $D_4$ x 168, $C_2^3$ x 113, $Q_8$ x 40

Order 4: $C_2^2$ x 111, $C_4$ x 34

Order 2: $C_2$ x 25

Order 1: $C_1$

Classes of subgroups up to automorphism

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

Order 128: $C_4.C_2^5$, $C_4^2:C_2^3$, $C_2^3.C_2^4$, $C_2^4.D_4$, $C_2^4:D_4$, $C_2^3.C_2^4$

Order 64: $C_4.C_2^4$ x 2, $D_4:C_2^3$ x 2, $D_4:C_2^3$ x 2, $C_4^2:C_2^2$ x 2, $C_2^3.C_2^3$, $C_2^4:C_4$, $C_4.C_2^4$, $C_2^4\times C_4$, $C_2^3.D_4$, $C_2^3:Q_8$, $C_2^3:D_4$, $C_2^3:D_4$, $C_4^2:C_2^2$, $C_4^2:C_2^2$, $C_2^3.D_4$, $C_2\wr C_2^2$

Order 32: $D_4:C_2^2$ x 10, $C_2^3\times C_4$ x 8, $C_2^2\times D_4$ x 5, $D_4:C_2^2$ x 4, $C_2^3:C_4$ x 3, $C_4.C_2^3$ x 2, $C_2^2\times Q_8$ x 2, $C_2^2.D_4$ x 2, $C_2^2\wr C_2$ x 2, $C_4^2:C_2$ x 2, $C_2^2.D_4$ x 2, $C_2^3:C_4$, $C_2^5$, $C_2^2:Q_8$, $C_4:D_4$, $C_4\times D_4$, $C_2\times C_4^2$

Order 16: $C_2^2\times C_4$ x 21, $C_2\times D_4$ x 13, $D_4:C_2$ x 8, $C_2^4$ x 7, $C_2\times Q_8$ x 5, $C_2^2:C_4$ x 4, $C_4:C_4$ x 2, $C_4^2$

Order 8: $C_2\times C_4$ x 19, $C_2^3$ x 17, $D_4$ x 5, $Q_8$ x 2

Order 4: $C_2^2$ x 16, $C_4$ x 5

Order 2: $C_2$ x 6

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 16, $C_2^4.D_4$ ($C_2$) x 4, $C_2^4:D_4$ ($C_2$) x 4, $C_4^2:C_2^3$ ($C_2$) x 3, $C_2^3.C_2^4$ ($C_2$) x 3, $C_4.C_2^5$ ($C_2$)

Order 64: $C_2^3.D_4$ ($C_2^2$) x 32, $C_2\wr C_2^2$ ($C_2^2$) x 32, $C_2^3.C_2^3$ ($C_2^2$) x 24, $C_4^2:C_2^2$ ($C_2^2$) x 24, $C_2^4:C_4$ ($C_2^2$) x 12, $C_4.C_2^4$ ($C_2^2$) x 8, $C_2^3.D_4$ ($C_2^2$) x 6, $C_2^3:D_4$ ($C_2^2$) x 6, $D_4:C_2^3$ ($C_2^2$) x 4, $D_4:C_2^3$ ($C_2^2$) x 3, $C_4^2:C_2^2$ ($C_2^2$) x 3, $C_2^4\times C_4$ ($C_2^2$)

Order 32: $C_2^3:C_4$ ($C_2^3$) x 48, $C_2^2.D_4$ ($C_2^3$) x 24, $C_2^2\wr C_2$ ($C_2^3$) x 24, $D_4:C_2^2$ ($C_2^3$) x 16, $D_4:C_2^2$ ($C_2^3$) x 12, $D_4:C_2^2$ ($D_4$) x 12, $C_4^2:C_2$ ($C_2^3$) x 12, $C_2^2\times D_4$ ($C_2^3$) x 6, $C_2^3\times C_4$ ($C_2^3$) x 6, $C_2^3\times C_4$ ($D_4$) x 6, $C_2^3:C_4$ ($C_2^3$) x 6, $C_2^2\times Q_8$ ($D_4$) x 3, $C_2^2\times D_4$ ($D_4$) x 3, $C_2^5$ ($C_2^3$)

Order 16: $C_2^2\times C_4$ ($C_2\times D_4$) x 42, $C_2\times D_4$ ($C_2\times D_4$) x 24, $C_2^2:C_4$ ($C_2^4$) x 12, $C_2\times Q_8$ ($C_2\times D_4$) x 12, $C_2\times D_4$ ($C_2^4$) x 12, $C_2^4$ ($C_2\times D_4$) x 6, $C_2^2\times C_4$ ($C_2^4$) x 4, $C_2^4$ ($C_2^4$) x 3

Order 8: $C_2\times C_4$ ($C_2^2\times D_4$) x 24, $C_2^3$ ($C_2^2\times D_4$) x 18, $C_2\times C_4$ ($C_2^2\wr C_2$) x 16, $C_2^3$ ($C_2^5$)

Order 4: $C_4$ ($C_2^3:D_4$) x 8, $C_2^2$ ($C_2^3:D_4$) x 4, $C_2^2$ ($D_4\times C_2^3$) x 3

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

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: $C_4.C_2^5$ ($C_2$), $C_4^2:C_2^3$ ($C_2$), $C_2^3.C_2^4$ ($C_2$), $C_2^4.D_4$ ($C_2$), $C_2^4:D_4$ ($C_2$), $C_2^3.C_2^4$ ($C_2$)

Order 64: $C_2^3.C_2^3$ ($C_2^2$), $C_2^4:C_4$ ($C_2^2$), $C_4.C_2^4$ ($C_2^2$), $D_4:C_2^3$ ($C_2^2$), $D_4:C_2^3$ ($C_2^2$), $C_2^4\times C_4$ ($C_2^2$), $C_4^2:C_2^2$ ($C_2^2$), $C_2^3.D_4$ ($C_2^2$), $C_2^3:D_4$ ($C_2^2$), $C_4^2:C_2^2$ ($C_2^2$), $C_2^3.D_4$ ($C_2^2$), $C_2\wr C_2^2$ ($C_2^2$)

Order 32: $C_2^3\times C_4$ ($C_2^3$) x 3, $C_2^3:C_4$ ($C_2^3$), $C_2^5$ ($C_2^3$), $D_4:C_2^2$ ($C_2^3$), $D_4:C_2^2$ ($C_2^3$), $D_4:C_2^2$ ($D_4$), $C_2^2\times Q_8$ ($D_4$), $C_2^2\times D_4$ ($C_2^3$), $C_2^2\times D_4$ ($D_4$), $C_2^3\times C_4$ ($D_4$), $C_2^2.D_4$ ($C_2^3$), $C_2^2\wr C_2$ ($C_2^3$), $C_4^2:C_2$ ($C_2^3$), $C_2^3:C_4$ ($C_2^3$)

Order 16: $C_2^2\times C_4$ ($C_2\times D_4$) x 5, $C_2^4$ ($C_2^4$) x 2, $C_2\times D_4$ ($C_2\times D_4$) x 2, $C_2^2\times C_4$ ($C_2^4$) x 2, $C_2^2:C_4$ ($C_2^4$), $C_2^4$ ($C_2\times D_4$), $C_2\times Q_8$ ($C_2\times D_4$), $C_2\times D_4$ ($C_2^4$)

Order 8: $C_2\times C_4$ ($C_2^2\wr C_2$) x 4, $C_2^3$ ($C_2^2\times D_4$) x 3, $C_2\times C_4$ ($C_2^2\times D_4$) x 3, $C_2^3$ ($C_2^5$)

Order 4: $C_2^2$ ($C_2^3:D_4$) x 2, $C_4$ ($C_2^3:D_4$) x 2, $C_2^2$ ($D_4\times C_2^3$)

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

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

$C_2^3.C_2^5$

$\rhd$

$C_2^3.C_2^4$

$\rhd$

$C_2\wr C_2^2$

$\rhd$

$C_2^2\wr C_2$

$\rhd$

$C_2^4$

$\rhd$

$C_2^3$

$\rhd$

$C_2^2$

$\rhd$

$C_2$

$\rhd$

$C_1$

Lower central series

$C_2^3.C_2^5$

$\rhd$

$C_2^3$

$\rhd$

$C_2$

$\rhd$

$C_1$

Upper central series

$C_1$

$\lhd$

$C_2\times C_4$

$\lhd$

$C_2^3\times C_4$

$\lhd$

$C_2^3.C_2^5$

Supergroups

This group is a maximal subgroup of 11 larger groups in the database.

This group is a maximal quotient of 7 larger groups in the database.

Character theory

Complex character table

See the $64 \times 64$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $60 \times 60$ rational character table.