Abstract group 512.10494202: $D_4\times C_2^6$

• Label: 512.10494202
• Order: \( 2^{9} \)
• Exponent: \( 2^{2} \)
• Nilpotent: yes
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{8} \)
• $\card{Z(G)}$: \( 2^{7} \)
• $\card{\Aut(G)}$: \( 2^{36} \cdot 3^{4} \cdot 5 \cdot 7^{2} \cdot 31 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{34} \cdot 3^{4} \cdot 5 \cdot 7^{2} \cdot 31 \)
• Perm deg.: not computed
• Trans deg.: $256$
• Rank: $8$

Group information

Description:	$D_4\times C_2^6$
Order:	\(512\)\(\medspace = 2^{9} \)
Exponent:	\(4\)\(\medspace = 2^{2} \)
Automorphism group:	$C_2^8.C_2^6.C_2.C_2^6.\GL(6,2)$, of order \(42275878490603520\)\(\medspace = 2^{36} \cdot 3^{4} \cdot 5 \cdot 7^{2} \cdot 31 \)
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	383	128	512
Conjugacy classes	1	255	64	320
Divisions	1	255	64	320
Autjugacy classes	1	3	1	5

Dimension	1	2
Irr. complex chars.	256	64	320
Irr. rational chars.	256	64	320

Minimal presentations

Permutation degree:	not computed
Transitive degree:	$256$
Rank:	$8$
Inequivalent generating 8-tuples:	$32385$

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:	${\langle a, b, c, d, e, f, g, h \mid a^{2}=b^{4}=c^{2}=d^{2}=e^{2}=f^{2}= \!\cdots\! \rangle}$
Permutation group:	Degree $256$ $\langle(1,109)(2,110)(3,111)(4,112)(5,105)(6,106)(7,107)(8,108)(9,101)(10,102) \!\cdots\! \rangle$
Direct product:	$C_2$ ${}^6$ $\, \times\, $ $D_4$
Semidirect product:	$C_2^8$ $\,\rtimes\,$ $C_2$	$C_4$ $\,\rtimes\,$ $C_2^7$	$C_2^7$ $\,\rtimes\,$ $C_2^2$	$C_2^6$ $\,\rtimes\,$ $C_2^3$	all 14
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$C_2$ . $C_2^8$	$C_2^7$ . $C_2^2$	$C_2^6$ . $C_2^3$	$C_2^5$ . $C_2^4$	all 7
Aut. group:	$\Aut(C_8\times \SD_{16})$	$\Aut(\SD_{16}:C_8)$	$\Aut(C_8:D_8)$	$\Aut(C_8:\SD_{16})$	all 28

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

Homology

Abelianization:	$C_{2}^{8} $
Schur multiplier:	$C_{2}^{28}$
Commutator length:	$1$

Subgroups

There are 1166786 subgroups in 805186 conjugacy classes, 443586 normal (5 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_2^7$	$G/Z \simeq$ $C_2^2$
Commutator:	$G' \simeq$ $C_2$	$G/G' \simeq$ $C_2^8$
Frattini:	$\Phi \simeq$ $C_2$	$G/\Phi \simeq$ $C_2^8$
Fitting:	$\operatorname{Fit} \simeq$ $D_4\times C_2^6$	$G/\operatorname{Fit} \simeq$ $C_1$
Radical:	$R \simeq$ $D_4\times C_2^6$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_2^7$	$G/\operatorname{soc} \simeq$ $C_2^2$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $D_4\times C_2^6$

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

Normal subgroups

No diagram available: subgroups only stored up to automorphism

Normal subgroups up to automorphism

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Classes of subgroups up to conjugation

Order 512: $D_4\times C_2^6$

Order 256: $D_4\times C_2^5$ x 252, $C_2^8$ x 2, $C_2^6\times C_4$

Order 128: $D_4\times C_2^4$ x 10416, $C_2^7$ x 381, $C_2^5\times C_4$ x 126

Order 64: $D_4\times C_2^3$ x 89280, $C_2^6$ x 13399, $C_2^4\times C_4$ x 2604

Order 32: $C_2^2\times D_4$ x 166656, $C_2^5$ x 108315, $C_2^3\times C_4$ x 11160

Order 16: $C_2^4$ x 211203, $C_2\times D_4$ x 64512, $C_2^2\times C_4$ x 10416

Order 8: $C_2^3$ x 99171, $D_4$ x 4096, $C_2\times C_4$ x 2016

Order 4: $C_2^2$ x 10859, $C_4$ x 64

Order 2: $C_2$ x 255

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 512: $D_4\times C_2^6$

Order 256: $C_2^8$, $D_4\times C_2^5$, $C_2^6\times C_4$

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

Order 64: $C_2^6$ x 4, $D_4\times C_2^3$, $C_2^4\times C_4$

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

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

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

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

Order 2: $C_2$ x 3

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 512: $D_4\times C_2^6$ ($C_1$)

Order 256: $D_4\times C_2^5$ ($C_2$) x 252, $C_2^8$ ($C_2$) x 2, $C_2^6\times C_4$ ($C_2$)

Order 128: $D_4\times C_2^4$ ($C_2^2$) x 10416, $C_2^7$ ($C_2^2$) x 253, $C_2^5\times C_4$ ($C_2^2$) x 126

Order 64: $D_4\times C_2^3$ ($C_2^3$) x 89280, $C_2^6$ ($C_2^3$) x 5271, $C_2^4\times C_4$ ($C_2^3$) x 2604, $C_2^6$ ($D_4$) x 64

Order 32: $C_2^2\times D_4$ ($C_2^4$) x 166656, $C_2^5$ ($C_2^4$) x 22971, $C_2^3\times C_4$ ($C_2^4$) x 11160, $C_2^5$ ($C_2\times D_4$) x 2016

Order 16: $C_2\times D_4$ ($C_2^5$) x 64512, $C_2^4$ ($C_2^5$) x 22227, $C_2^4$ ($C_2^2\times D_4$) x 10416, $C_2^2\times C_4$ ($C_2^5$) x 10416

Order 8: $C_2^3$ ($D_4\times C_2^3$) x 11160, $C_2^3$ ($C_2^6$) x 4683, $D_4$ ($C_2^6$) x 4096, $C_2\times C_4$ ($C_2^6$) x 2016

Order 4: $C_2^2$ ($D_4\times C_2^4$) x 2604, $C_2^2$ ($C_2^7$) x 191, $C_4$ ($C_2^7$) x 64

Order 2: $C_2$ ($D_4\times C_2^5$) x 126, $C_2$ ($C_2^8$)

Order 1: $C_1$ ($D_4\times C_2^6$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 512: $D_4\times C_2^6$ ($C_1$)

Order 256: $C_2^8$ ($C_2$), $D_4\times C_2^5$ ($C_2$), $C_2^6\times C_4$ ($C_2$)

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

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

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

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

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

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

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

Order 1: $C_1$ ($D_4\times C_2^6$)

Series

Derived series

$D_4\times C_2^6$

$\rhd$

$C_2$

$\rhd$

$C_1$

Chief series

$D_4\times C_2^6$

$\rhd$

$C_2^6\times C_4$

$\rhd$

$C_2^7$

$\rhd$

$C_2^6$

$\rhd$

$C_2^5$

$\rhd$

$C_2^4$

$\rhd$

$C_2^3$

$\rhd$

$C_2^2$

$\rhd$

$C_2$

$\rhd$

$C_1$

Lower central series

$D_4\times C_2^6$

$\rhd$

$C_2$

$\rhd$

$C_1$

Upper central series

$C_1$

$\lhd$

$C_2^7$

$\lhd$

$D_4\times C_2^6$

Supergroups

This group is a maximal subgroup of 12 larger groups in the database.

This group is a maximal quotient of 4 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 $320 \times 320$ rational character table (warning: may be slow to load). Alternatively, you may search for characters of this group with desired properties.