Abstract group 512.7572924: $C_2^3:D_4^2$

• Label: 512.7572924
• Order: \( 2^{9} \)
• Exponent: \( 2^{2} \)
• Nilpotent: yes
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{6} \)
• $\card{Z(G)}$: \( 2^{4} \)
• $\card{\Aut(G)}$: \( 2^{24} \cdot 3 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{19} \cdot 3 \)
• Perm deg.: not computed
• Trans deg.: $64$
• Rank: $6$

Group information

Description:	$C_2^3:D_4^2$
Order:	\(512\)\(\medspace = 2^{9} \)
Exponent:	\(4\)\(\medspace = 2^{2} \)
Automorphism group:	$C_2^{18}.(D_4\times S_4)$, of order \(50331648\)\(\medspace = 2^{24} \cdot 3 \)
Composition factors:	$C_2$ x 9
Nilpotency class:	$2$
Derived length:	$2$

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

Group statistics

Order	1	2	4
Elements	1	223	288	512
Conjugacy classes	1	73	54	128
Divisions	1	73	54	128
Autjugacy classes	1	9	4	14

Dimension	1	2	4
Irr. complex chars.	64	48	16	128
Irr. rational chars.	64	48	16	128

Minimal presentations

Permutation degree:	not computed
Transitive degree:	$64$
Rank:	$6$
Inequivalent generating 6-tuples:	$104993280$

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:	${\langle a, b, c, d, e, f, g \mid a^{2}=b^{4}=c^{2}=d^{4}=e^{2}=f^{2}=g^{2}= \!\cdots\! \rangle}$
Direct product:	$C_2$ $\, \times\, $ $(C_2^2:D_4^2)$
Semidirect product:	$C_2^3$ $\,\rtimes\,$ $D_4^2$	$C_2^7$ $\,\rtimes\,$ $C_2^2$	$C_2^6$ $\,\rtimes\,$ $C_2^3$ (2)	$C_2^5$ $\,\rtimes\,$ $C_2^4$	all 70
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$C_2^6$ . $C_2^3$ (4)	$C_2^5$ . $C_2^4$ (9)	$C_2^4$ . $C_2^5$ (6)	$C_2^3$ . $C_2^6$	all 27

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

Homology

Abelianization:	$C_{2}^{6} $
Schur multiplier:	$C_{2}^{15}$
Commutator length:	$1$

Subgroups

There are 74402 subgroups in 25264 conjugacy classes, 3906 normal (14 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_2^4$	$G/Z \simeq$ $C_2^5$
Commutator:	$G' \simeq$ $C_2^3$	$G/G' \simeq$ $C_2^6$
Frattini:	$\Phi \simeq$ $C_2^3$	$G/\Phi \simeq$ $C_2^6$
Fitting:	$\operatorname{Fit} \simeq$ $C_2^3:D_4^2$	$G/\operatorname{Fit} \simeq$ $C_1$
Radical:	$R \simeq$ $C_2^3:D_4^2$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_2^4$	$G/\operatorname{soc} \simeq$ $C_2^5$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2^3:D_4^2$

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 512: $C_2^3:D_4^2$

Order 256: $C_2^2:D_4^2$ x 32, $C_2^2.D_4^2$ x 6, $C_2^5:D_4$ x 6, $C_2^2\times D_4^2$ x 3, $C_2^5:D_4$ x 3, $C_2^2.D_4^2$ x 3, $C_2^2.D_4^2$ x 3, $C_2^5:D_4$ x 3, $C_2^2.D_4^2$ x 3, $C_2^5:C_2^3$

Order 128: $C_2^4:D_4$ x 130, $C_2\times D_4^2$ x 96, $C_2.D_4^2$ x 96, $C_2^4:D_4$ x 96, $C_2.D_4^2$ x 48, $C_2.D_4^2$ x 48, $C_2^4:D_4$ x 48, $C_2.D_4^2$ x 48, $C_2^4:D_4$ x 18, $C_2^4.D_4$ x 18, $C_2.D_4^2$ x 16, $C_2^4:D_4$ x 15, $D_4\times C_2^4$ x 12, $C_2^4.D_4$ x 12, $C_2^5:C_4$ x 9, $C_4^2:C_2^3$ x 6, $C_2^3.C_2^4$ x 6, $C_2^3.C_2^4$ x 6, $C_2^4.D_4$ x 6, $C_4^2:C_2^3$ x 4, $C_2^5:C_4$ x 3, $C_2^3:C_4^2$ x 3, $C_2^4.Q_8$ x 2, $C_2^7$

Order 64: $C_2^3:D_4$ x 948, $D_4^2$ x 384, $D_4\times C_2^3$ x 369, $C_2^3:D_4$ x 228, $C_2^4:C_4$ x 204, $C_2^3:D_4$ x 144, $C_2^3.D_4$ x 144, $C_2^3.D_4$ x 96, $C_4^2:C_2^2$ x 96, $C_4^2:C_2^2$ x 60, $C_2^6$ x 59, $C_2^3.C_2^3$ x 48, $C_2^3.C_2^3$ x 48, $C_2^3.D_4$ x 48, $C_2^4:C_4$ x 24, $C_2^2:C_4^2$ x 24, $C_2^3.Q_8$ x 16, $C_2^2.C_4^2$ x 14, $C_2^4\times C_4$ x 12, $C_2^3.D_4$ x 9, $C_2^2\times C_4^2$ x 4

Order 32: $C_2^2\times D_4$ x 2352, $C_2^2\wr C_2$ x 1168, $C_2^5$ x 927, $C_2^3:C_4$ x 786, $C_4:D_4$ x 432, $C_4\times D_4$ x 192, $C_2^3\times C_4$ x 186, $C_4:D_4$ x 112, $C_2^2.D_4$ x 66, $C_2.C_4^2$ x 56, $C_2\times C_4^2$ x 30

Order 16: $C_2^4$ x 3957, $C_2\times D_4$ x 3312, $C_2^2\times C_4$ x 594, $C_2^2:C_4$ x 516, $C_4:C_4$ x 60, $C_4^2$ x 28

Order 8: $C_2^3$ x 4233, $D_4$ x 792, $C_2\times C_4$ x 426

Order 4: $C_2^2$ x 1121, $C_4$ x 54

Order 2: $C_2$ x 73

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 512: $C_2^3:D_4^2$

Order 256: $C_2^2\times D_4^2$, $C_2^5:D_4$, $C_2^2:D_4^2$, $C_2^5:C_2^3$, $C_2^2.D_4^2$, $C_2^2.D_4^2$, $C_2^2.D_4^2$, $C_2^5:D_4$, $C_2^5:D_4$, $C_2^2.D_4^2$

Order 128: $C_2^4:D_4$ x 11, $C_2^4.D_4$ x 4, $D_4\times C_2^4$ x 3, $C_2^4:D_4$ x 3, $C_2^4:D_4$ x 3, $C_2\times D_4^2$ x 2, $C_4^2:C_2^3$ x 2, $C_2^5:C_4$ x 2, $C_2^4.D_4$ x 2, $C_2^3.C_2^4$ x 2, $C_2^7$, $C_4^2:C_2^3$, $C_2.D_4^2$, $C_2.D_4^2$, $C_2.D_4^2$, $C_2.D_4^2$, $C_2^4:D_4$, $C_2^4:D_4$, $C_2^4.Q_8$, $C_2.D_4^2$, $C_2^3.C_2^4$, $C_2^4.D_4$, $C_2^5:C_4$, $C_2^3:C_4^2$

Order 64: $D_4\times C_2^3$ x 21, $C_2^3:D_4$ x 19, $C_2^4:C_4$ x 16, $C_2^6$ x 9, $C_2^3:D_4$ x 6, $C_2^3.D_4$ x 4, $C_4^2:C_2^2$ x 4, $C_2^3:D_4$ x 3, $C_2^2.C_4^2$ x 3, $C_2^4\times C_4$ x 3, $C_2^3.D_4$ x 2, $C_2^3.C_2^3$ x 2, $C_4^2:C_2^2$ x 2, $C_2^3.D_4$ x 2, $C_2^2\times C_4^2$ x 2, $C_2^3.Q_8$, $C_2^3.C_2^3$, $C_2^3.D_4$, $C_2^4:C_4$, $C_2^2:C_4^2$, $D_4^2$

Order 32: $C_2^2\times D_4$ x 51, $C_2^5$ x 37, $C_2^3:C_4$ x 26, $C_2^3\times C_4$ x 15, $C_2^2\wr C_2$ x 9, $C_2^2.D_4$ x 4, $C_2\times C_4^2$ x 4, $C_4:D_4$ x 3, $C_2.C_4^2$ x 3, $C_4:D_4$ x 2, $C_4\times D_4$

Order 16: $C_2^4$ x 75, $C_2\times D_4$ x 43, $C_2^2\times C_4$ x 27, $C_2^2:C_4$ x 12, $C_4:C_4$ x 2, $C_4^2$ x 2

Order 8: $C_2^3$ x 77, $C_2\times C_4$ x 19, $D_4$ x 10

Order 4: $C_2^2$ x 39, $C_4$ x 4

Order 2: $C_2$ x 9

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 512: $C_2^3:D_4^2$ ($C_1$)

Order 256: $C_2^2:D_4^2$ ($C_2$) x 32, $C_2^2.D_4^2$ ($C_2$) x 6, $C_2^5:D_4$ ($C_2$) x 6, $C_2^2\times D_4^2$ ($C_2$) x 3, $C_2^5:D_4$ ($C_2$) x 3, $C_2^2.D_4^2$ ($C_2$) x 3, $C_2^2.D_4^2$ ($C_2$) x 3, $C_2^5:D_4$ ($C_2$) x 3, $C_2^2.D_4^2$ ($C_2$) x 3, $C_2^5:C_2^3$ ($C_2$)

Order 128: $C_2.D_4^2$ ($C_2^2$) x 96, $C_2^4:D_4$ ($C_2^2$) x 96, $C_2^4:D_4$ ($C_2^2$) x 82, $C_2\times D_4^2$ ($C_2^2$) x 48, $C_2.D_4^2$ ($C_2^2$) x 48, $C_2.D_4^2$ ($C_2^2$) x 48, $C_2^4:D_4$ ($C_2^2$) x 48, $C_2.D_4^2$ ($C_2^2$) x 48, $C_2^4:D_4$ ($C_2^2$) x 18, $C_2^4.D_4$ ($C_2^2$) x 18, $C_2.D_4^2$ ($C_2^2$) x 16, $C_2^4:D_4$ ($C_2^2$) x 15, $D_4\times C_2^4$ ($C_2^2$) x 12, $C_2^4.D_4$ ($C_2^2$) x 12, $C_2^5:C_4$ ($C_2^2$) x 9, $C_4^2:C_2^3$ ($C_2^2$) x 6, $C_2^3.C_2^4$ ($C_2^2$) x 6, $C_2^3.C_2^4$ ($C_2^2$) x 6, $C_2^4.D_4$ ($C_2^2$) x 6, $C_4^2:C_2^3$ ($C_2^2$) x 4, $C_2^5:C_4$ ($C_2^2$) x 3, $C_2^3:C_4^2$ ($C_2^2$) x 3, $C_2^4.Q_8$ ($C_2^2$) x 2, $C_2^7$ ($C_2^2$)

Order 64: $C_2^3:D_4$ ($C_2^3$) x 272, $C_2^3:D_4$ ($C_2^3$) x 144, $C_2^3.D_4$ ($C_2^3$) x 144, $D_4\times C_2^3$ ($C_2^3$) x 141, $C_2^4:C_4$ ($C_2^3$) x 132, $C_2^3:D_4$ ($C_2^3$) x 120, $C_2^3.D_4$ ($C_2^3$) x 96, $C_2^3.C_2^3$ ($C_2^3$) x 48, $C_2^3.C_2^3$ ($C_2^3$) x 48, $C_2^3.D_4$ ($C_2^3$) x 48, $C_2^3:D_4$ ($D_4$) x 48, $C_4^2:C_2^2$ ($C_2^3$) x 48, $C_4^2:C_2^2$ ($C_2^3$) x 32, $C_2^4:C_4$ ($C_2^3$) x 24, $C_2^2:C_4^2$ ($C_2^3$) x 24, $C_2^6$ ($C_2^3$) x 19, $C_2^3.Q_8$ ($C_2^3$) x 16, $C_2^2.C_4^2$ ($C_2^3$) x 14, $C_2^4\times C_4$ ($C_2^3$) x 12, $C_2^3.D_4$ ($C_2^3$) x 9, $C_2^2\times C_4^2$ ($C_2^3$) x 4

Order 32: $C_2^3:C_4$ ($C_2^4$) x 240, $C_2^2\wr C_2$ ($C_2\times D_4$) x 192, $C_2^2\times D_4$ ($C_2^4$) x 180, $C_2^2\times D_4$ ($C_2\times D_4$) x 72, $C_2^3:C_4$ ($C_2\times D_4$) x 72, $C_2^3\times C_4$ ($C_2^4$) x 66, $C_2^5$ ($C_2^4$) x 57, $C_2.C_4^2$ ($C_2^4$) x 56, $C_2^2.D_4$ ($C_2^4$) x 36, $C_2^5$ ($C_2\times D_4$) x 24, $C_2\times C_4^2$ ($C_2^4$) x 16

Order 16: $C_2^2:C_4$ ($C_2^2\times D_4$) x 144, $C_2\times D_4$ ($C_2^2\times D_4$) x 144, $C_2^4$ ($C_2^2\times D_4$) x 96, $C_2^2\times C_4$ ($C_2^5$) x 36, $C_2^2\times C_4$ ($C_2^2\times D_4$) x 36, $C_2^4$ ($C_2^5$) x 27, $C_2^4$ ($D_4:C_2^2$) x 16

Order 8: $C_2^3$ ($D_4\times C_2^3$) x 54, $C_2^3$ ($D_4^2$) x 48, $C_2\times C_4$ ($D_4\times C_2^3$) x 36, $C_2^3$ ($D_4:C_2^3$) x 24, $C_2^3$ ($C_2^6$)

Order 4: $C_2^2$ ($C_2\times D_4^2$) x 36, $C_2^2$ ($D_4^2:C_2$) x 12, $C_2^2$ ($D_4:C_2^4$) x 4, $C_2^2$ ($D_4^2:C_2$) x 4, $C_2^2$ ($D_4\times C_2^4$) x 3

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

Order 1: $C_1$ ($C_2^3:D_4^2$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 512: $C_2^3:D_4^2$ ($C_1$)

Order 256: $C_2^2\times D_4^2$ ($C_2$), $C_2^5:D_4$ ($C_2$), $C_2^2:D_4^2$ ($C_2$), $C_2^5:C_2^3$ ($C_2$), $C_2^2.D_4^2$ ($C_2$), $C_2^2.D_4^2$ ($C_2$), $C_2^2.D_4^2$ ($C_2$), $C_2^5:D_4$ ($C_2$), $C_2^5:D_4$ ($C_2$), $C_2^2.D_4^2$ ($C_2$)

Order 128: $C_2^4:D_4$ ($C_2^2$) x 10, $C_2^4.D_4$ ($C_2^2$) x 4, $D_4\times C_2^4$ ($C_2^2$) x 3, $C_2^4:D_4$ ($C_2^2$) x 3, $C_2^4:D_4$ ($C_2^2$) x 3, $C_4^2:C_2^3$ ($C_2^2$) x 2, $C_2^5:C_4$ ($C_2^2$) x 2, $C_2^4.D_4$ ($C_2^2$) x 2, $C_2^3.C_2^4$ ($C_2^2$) x 2, $C_2^7$ ($C_2^2$), $C_2\times D_4^2$ ($C_2^2$), $C_4^2:C_2^3$ ($C_2^2$), $C_2.D_4^2$ ($C_2^2$), $C_2.D_4^2$ ($C_2^2$), $C_2.D_4^2$ ($C_2^2$), $C_2.D_4^2$ ($C_2^2$), $C_2^4:D_4$ ($C_2^2$), $C_2^4:D_4$ ($C_2^2$), $C_2^4.Q_8$ ($C_2^2$), $C_2.D_4^2$ ($C_2^2$), $C_2^3.C_2^4$ ($C_2^2$), $C_2^4.D_4$ ($C_2^2$), $C_2^5:C_4$ ($C_2^2$), $C_2^3:C_4^2$ ($C_2^2$)

Order 64: $C_2^4:C_4$ ($C_2^3$) x 14, $D_4\times C_2^3$ ($C_2^3$) x 13, $C_2^3:D_4$ ($C_2^3$) x 9, $C_2^6$ ($C_2^3$) x 6, $C_2^3.D_4$ ($C_2^3$) x 4, $C_2^3:D_4$ ($C_2^3$) x 3, $C_2^2.C_4^2$ ($C_2^3$) x 3, $C_2^4\times C_4$ ($C_2^3$) x 3, $C_2^3:D_4$ ($C_2^3$) x 3, $C_2^3.D_4$ ($C_2^3$) x 2, $C_2^3.C_2^3$ ($C_2^3$) x 2, $C_4^2:C_2^2$ ($C_2^3$) x 2, $C_2^3.D_4$ ($C_2^3$) x 2, $C_2^2\times C_4^2$ ($C_2^3$) x 2, $C_2^3.Q_8$ ($C_2^3$), $C_2^3.C_2^3$ ($C_2^3$), $C_2^3.D_4$ ($C_2^3$), $C_2^4:C_4$ ($C_2^3$), $C_2^2:C_4^2$ ($C_2^3$), $C_2^3:D_4$ ($D_4$), $C_4^2:C_2^2$ ($C_2^3$)

Order 32: $C_2^3:C_4$ ($C_2^4$) x 12, $C_2^5$ ($C_2^4$) x 10, $C_2^2\times D_4$ ($C_2^4$) x 10, $C_2^3\times C_4$ ($C_2^4$) x 7, $C_2.C_4^2$ ($C_2^4$) x 3, $C_2^2\times D_4$ ($C_2\times D_4$) x 2, $C_2^2.D_4$ ($C_2^4$) x 2, $C_2^3:C_4$ ($C_2\times D_4$) x 2, $C_2\times C_4^2$ ($C_2^4$) x 2, $C_2^5$ ($C_2\times D_4$), $C_2^2\wr C_2$ ($C_2\times D_4$)

Order 16: $C_2^4$ ($C_2^5$) x 6, $C_2^4$ ($C_2^2\times D_4$) x 4, $C_2^2\times C_4$ ($C_2^5$) x 4, $C_2^2:C_4$ ($C_2^2\times D_4$) x 2, $C_2^4$ ($D_4:C_2^2$) x 2, $C_2\times D_4$ ($C_2^2\times D_4$) x 2, $C_2^2\times C_4$ ($C_2^2\times D_4$) x 2

Order 8: $C_2^3$ ($D_4:C_2^3$) x 4, $C_2^3$ ($D_4\times C_2^3$) x 4, $C_2\times C_4$ ($D_4\times C_2^3$) x 2, $C_2^3$ ($C_2^6$), $C_2^3$ ($D_4^2$)

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

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

Order 1: $C_1$ ($C_2^3:D_4^2$)

Series

Derived series

$C_2^3:D_4^2$

$\rhd$

$C_2^3$

$\rhd$

$C_1$

Chief series

$C_2^3:D_4^2$

$\rhd$

$C_2^2.D_4^2$

$\rhd$

$C_2^4:D_4$

$\rhd$

$C_2^4:C_4$

$\rhd$

$C_2^5$

$\rhd$

$C_2^4$

$\rhd$

$C_2^3$

$\rhd$

$C_2^2$

$\rhd$

$C_2$

$\rhd$

$C_1$

Lower central series

$C_2^3:D_4^2$

$\rhd$

$C_2^3$

$\rhd$

$C_1$

Upper central series

$C_1$

$\lhd$

$C_2^4$

$\lhd$

$C_2^3:D_4^2$

Supergroups

This group is a maximal subgroup of 8 larger groups in the database.

This group is a maximal quotient of 3 larger groups in the database.

Character theory

Complex character table

Every character has rational values, so the complex character table is the same as the rational character table below.

Rational character table

See the $128 \times 128$ rational character table (warning: may be slow to load). Alternatively, you may search for characters of this group with desired properties.