Abstract group 512.10493114: $C_2^3\times D_4^2$

• Label: 512.10493114
• Order: \( 2^{9} \)
• Exponent: \( 2^{2} \)
• Nilpotent: yes
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{7} \)
• $\card{Z(G)}$: \( 2^{5} \)
• $\card{\Aut(G)}$: \( 2^{32} \cdot 3 \cdot 7 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{28} \cdot 3 \cdot 7 \)
• Perm deg.: not computed
• Trans deg.: $128$
• Rank: $7$

Group information

Description:	$C_2^3\times D_4^2$
Order:	\(512\)\(\medspace = 2^{9} \)
Exponent:	\(4\)\(\medspace = 2^{2} \)
Automorphism group:	Group of order \(90194313216\)\(\medspace = 2^{32} \cdot 3 \cdot 7 \)
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	287	224	512
Conjugacy classes	1	127	72	200
Divisions	1	127	72	200
Autjugacy classes	1	5	3	9

Dimension	1	2	4
Irr. complex chars.	128	64	8	200
Irr. rational chars.	128	64	8	200

Minimal presentations

Permutation degree:	not computed
Transitive degree:	$128$
Rank:	$7$
Inequivalent generating 7-tuples:	$29763720$

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:	${\langle a, b, c, d, e, f, g \mid a^{2}=b^{4}=c^{2}=d^{4}=e^{2}=f^{2}=g^{2}= \!\cdots\! \rangle}$
Permutation group:	Degree $128$ $\langle(1,65)(2,66)(3,67)(4,68)(5,69)(6,70)(7,71)(8,72)(9,73)(10,74)(11,75)(12,76) \!\cdots\! \rangle$
Direct product:	$C_2$ ${}^3$ $\, \times\, $ $D_4$ ${}^2$
Semidirect product:	$C_2^7$ $\,\rtimes\,$ $C_2^2$	$C_2^6$ $\,\rtimes\,$ $C_2^3$	$C_2^5$ $\,\rtimes\,$ $C_2^4$	$C_2^4$ $\,\rtimes\,$ $C_2^5$	all 49
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$C_2^6$ . $C_2^3$ (2)	$C_2^5$ . $C_2^4$ (3)	$C_2^4$ . $C_2^5$ (3)	$C_2^3$ . $C_2^6$ (3)	all 21
Aut. group:	$\Aut(C_{12}.D_8)$

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

Homology

Abelianization:	$C_{2}^{7} $
Schur multiplier:	$C_{2}^{21}$
Commutator length:	$1$

Subgroups

There are 220802 subgroups in 107122 conjugacy classes, 34370 normal (8 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_2^5$	$G/Z \simeq$ $C_2^4$
Commutator:	$G' \simeq$ $C_2^2$	$G/G' \simeq$ $C_2^7$
Frattini:	$\Phi \simeq$ $C_2^2$	$G/\Phi \simeq$ $C_2^7$
Fitting:	$\operatorname{Fit} \simeq$ $C_2^3\times D_4^2$	$G/\operatorname{Fit} \simeq$ $C_1$
Radical:	$R \simeq$ $C_2^3\times D_4^2$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_2^5$	$G/\operatorname{soc} \simeq$ $C_2^4$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2^3\times D_4^2$

Hi

Subgroup diagram and profile

Classes of subgroups up to conjugation

No diagram available: subgroups only stored up to automorphism

Classes of subgroups up to automorphism

No diagram available

Normal subgroups

No diagram available: subgroups only stored up to automorphism

Normal subgroups up to automorphism

No diagram available

Classes of subgroups up to conjugation

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

Order 256: $C_2^2\times D_4^2$ x 112, $D_4\times C_2^5$ x 4, $C_2^5:D_4$ x 4, $C_2^5:D_4$ x 4, $C_4^2:C_2^4$ x 2, $C_4^2:C_2^4$

Order 128: $C_2\times D_4^2$ x 1792, $D_4\times C_2^4$ x 368, $C_2^4:D_4$ x 224, $C_2^4:D_4$ x 224, $C_4^2:C_2^3$ x 112, $C_4^2:C_2^3$ x 56, $C_2^5:C_4$ x 8, $C_2^7$ x 4, $C_2^5\times C_4$ x 4, $C_2^4.D_4$ x 2, $C_2^3\times C_4^2$

Order 64: $D_4\times C_2^3$ x 6096, $D_4^2$ x 4096, $C_2^3:D_4$ x 1792, $C_2^3:D_4$ x 1792, $C_4^2:C_2^2$ x 896, $C_4^2:C_2^2$ x 448, $C_2^6$ x 280, $C_2^4:C_4$ x 224, $C_2^4\times C_4$ x 183, $C_2^3.D_4$ x 56, $C_2^2\times C_4^2$ x 28

Order 32: $C_2^2\times D_4$ x 21984, $C_2^5$ x 4049, $C_4:D_4$ x 2048, $C_2^2\wr C_2$ x 2048, $C_2^3\times C_4$ x 1514, $C_4\times D_4$ x 1024, $C_2^3:C_4$ x 896, $C_4:D_4$ x 512, $C_2^2.D_4$ x 224, $C_2\times C_4^2$ x 112

Order 16: $C_2\times D_4$ x 16832, $C_2^4$ x 14495, $C_2^2\times C_4$ x 2748, $C_2^2:C_4$ x 512, $C_4:C_4$ x 128, $C_4^2$ x 64

Order 8: $C_2^3$ x 12923, $D_4$ x 2176, $C_2\times C_4$ x 1072

Order 4: $C_2^2$ x 2747, $C_4$ x 72

Order 2: $C_2$ x 127

Order 1: $C_1$

Classes of subgroups up to automorphism

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

Order 256: $D_4\times C_2^5$, $C_2^2\times D_4^2$, $C_4^2:C_2^4$, $C_2^5:D_4$, $C_2^5:D_4$, $C_4^2:C_2^4$

Order 128: $D_4\times C_2^4$ x 6, $C_2^5:C_4$ x 2, $C_2^7$, $C_2^5\times C_4$, $C_2\times D_4^2$, $C_4^2:C_2^3$, $C_2^4:D_4$, $C_2^4:D_4$, $C_4^2:C_2^3$, $C_2^4.D_4$, $C_2^3\times C_4^2$

Order 64: $D_4\times C_2^3$ x 10, $C_2^6$ x 5, $C_2^4\times C_4$ x 5, $C_2^4:C_4$ x 2, $D_4^2$, $C_4^2:C_2^2$, $C_2^3:D_4$, $C_2^3:D_4$, $C_4^2:C_2^2$, $C_2^3.D_4$, $C_2^2\times C_4^2$

Order 32: $C_2^5$ x 12, $C_2^2\times D_4$ x 10, $C_2^3\times C_4$ x 8, $C_2^3:C_4$ x 2, $C_4:D_4$, $C_4:D_4$, $C_2^2\wr C_2$, $C_4\times D_4$, $C_2^2.D_4$, $C_2\times C_4^2$

Order 16: $C_2^4$ x 16, $C_2\times D_4$ x 9, $C_2^2\times C_4$ x 8, $C_2^2:C_4$ x 2, $C_4:C_4$, $C_4^2$

Order 8: $C_2^3$ x 16, $C_2\times C_4$ x 7, $D_4$ x 4

Order 4: $C_2^2$ x 12, $C_4$ x 3

Order 2: $C_2$ x 5

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

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

Order 256: $C_2^2\times D_4^2$ ($C_2$) x 112, $D_4\times C_2^5$ ($C_2$) x 4, $C_2^5:D_4$ ($C_2$) x 4, $C_2^5:D_4$ ($C_2$) x 4, $C_4^2:C_2^4$ ($C_2$) x 2, $C_4^2:C_2^4$ ($C_2$)

Order 128: $C_2\times D_4^2$ ($C_2^2$) x 1792, $D_4\times C_2^4$ ($C_2^2$) x 240, $C_2^4:D_4$ ($C_2^2$) x 224, $C_2^4:D_4$ ($C_2^2$) x 224, $C_4^2:C_2^3$ ($C_2^2$) x 112, $C_4^2:C_2^3$ ($C_2^2$) x 56, $C_2^5:C_4$ ($C_2^2$) x 8, $C_2^7$ ($C_2^2$) x 4, $C_2^5\times C_4$ ($C_2^2$) x 4, $C_2^4.D_4$ ($C_2^2$) x 2, $C_2^3\times C_4^2$ ($C_2^2$)

Order 64: $D_4^2$ ($C_2^3$) x 4096, $D_4\times C_2^3$ ($C_2^3$) x 2240, $C_2^3:D_4$ ($C_2^3$) x 1792, $C_2^3:D_4$ ($C_2^3$) x 1792, $C_4^2:C_2^2$ ($C_2^3$) x 896, $C_4^2:C_2^2$ ($C_2^3$) x 448, $C_2^4:C_4$ ($C_2^3$) x 224, $C_2^6$ ($C_2^3$) x 120, $C_2^4\times C_4$ ($C_2^3$) x 119, $D_4\times C_2^3$ ($D_4$) x 64, $C_2^3.D_4$ ($C_2^3$) x 56, $C_2^2\times C_4^2$ ($C_2^3$) x 28

Order 32: $C_2^2\times D_4$ ($C_2^4$) x 3840, $C_4:D_4$ ($C_2^4$) x 2048, $C_2^2\wr C_2$ ($C_2^4$) x 2048, $C_4\times D_4$ ($C_2^4$) x 1024, $C_2^2\times D_4$ ($C_2\times D_4$) x 896, $C_2^3:C_4$ ($C_2^4$) x 896, $C_2^5$ ($C_2^4$) x 561, $C_2^3\times C_4$ ($C_2^4$) x 546, $C_4:D_4$ ($C_2^4$) x 512, $C_2^2.D_4$ ($C_2^4$) x 224, $C_2\times C_4^2$ ($C_2^4$) x 112, $C_2^5$ ($C_2\times D_4$) x 64, $C_2^3\times C_4$ ($C_2\times D_4$) x 32

Order 16: $C_2\times D_4$ ($C_2^2\times D_4$) x 1792, $C_2\times D_4$ ($C_2^5$) x 1024, $C_2^2:C_4$ ($C_2^5$) x 512, $C_2^4$ ($C_2^5$) x 487, $C_2^4$ ($C_2^2\times D_4$) x 464, $C_2^2\times C_4$ ($C_2^5$) x 452, $C_2^2\times C_4$ ($C_2^2\times D_4$) x 224, $C_4:C_4$ ($C_2^5$) x 128, $C_4^2$ ($C_2^5$) x 64, $C_2^4$ ($D_4:C_2^2$) x 8

Order 8: $D_4$ ($D_4\times C_2^3$) x 512, $C_2^3$ ($D_4\times C_2^3$) x 504, $C_2\times C_4$ ($D_4\times C_2^3$) x 224, $C_2^3$ ($C_2^6$) x 71, $C_2^3$ ($D_4^2$) x 64, $C_2\times C_4$ ($C_2^6$) x 56, $C_2^3$ ($D_4:C_2^3$) x 28

Order 4: $C_2^2$ ($C_2\times D_4^2$) x 112, $C_2^2$ ($D_4\times C_2^4$) x 92, $C_4$ ($D_4\times C_2^4$) x 32, $C_2^2$ ($D_4:C_2^4$) x 14, $C_2^2$ ($C_2^7$)

Order 2: $C_2$ ($C_2^2\times D_4^2$) x 28, $C_2$ ($D_4\times C_2^5$) x 2, $C_2$ ($D_4:C_2^5$)

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

Normal subgroups up to automorphism (quotient in parentheses)

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

Order 256: $D_4\times C_2^5$ ($C_2$), $C_2^2\times D_4^2$ ($C_2$), $C_4^2:C_2^4$ ($C_2$), $C_2^5:D_4$ ($C_2$), $C_2^5:D_4$ ($C_2$), $C_4^2:C_2^4$ ($C_2$)

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

Order 64: $D_4\times C_2^3$ ($C_2^3$) x 5, $C_2^4\times C_4$ ($C_2^3$) x 4, $C_2^6$ ($C_2^3$) x 3, $C_2^4:C_4$ ($C_2^3$) x 2, $D_4\times C_2^3$ ($D_4$), $D_4^2$ ($C_2^3$), $C_4^2:C_2^2$ ($C_2^3$), $C_2^3:D_4$ ($C_2^3$), $C_2^3:D_4$ ($C_2^3$), $C_4^2:C_2^2$ ($C_2^3$), $C_2^3.D_4$ ($C_2^3$), $C_2^2\times C_4^2$ ($C_2^3$)

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

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

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

Order 4: $C_2^2$ ($D_4\times C_2^4$) x 2, $C_2^2$ ($C_2^7$), $C_2^2$ ($D_4:C_2^4$), $C_2^2$ ($C_2\times D_4^2$), $C_4$ ($D_4\times C_2^4$)

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

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

Series

Derived series

$C_2^3\times D_4^2$

$\rhd$

$C_2^2$

$\rhd$

$C_1$

Chief series

$C_2^3\times D_4^2$

$\rhd$

$C_4^2:C_2^4$

$\rhd$

$D_4\times C_2^4$

$\rhd$

$C_2^4\times C_4$

$\rhd$

$C_2^5$

$\rhd$

$C_2^4$

$\rhd$

$C_2^3$

$\rhd$

$C_2^2$

$\rhd$

$C_2$

$\rhd$

$C_1$

Lower central series

$C_2^3\times D_4^2$

$\rhd$

$C_2^2$

$\rhd$

$C_1$

Upper central series

$C_1$

$\lhd$

$C_2^5$

$\lhd$

$C_2^3\times D_4^2$

Supergroups

This group is a maximal subgroup of 10 larger groups in the database.

This group is a maximal quotient of 6 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 $200 \times 200$ rational character table (warning: may be slow to load). Alternatively, you may search for characters of this group with desired properties.