Abstract group 512.6249624: $C_2^3\times C_8^2$

• Label: 512.6249624
• Order: \( 2^{9} \)
• Exponent: \( 2^{3} \)
• Abelian: yes
• $\card{\Aut(G)}$: \( 2^{24} \cdot 3^{2} \cdot 7 \)
• Perm deg.: $22$
• Trans deg.: $512$
• Rank: $5$

Group information

Description:	$C_2^3\times C_8^2$
Order:	\(512\)\(\medspace = 2^{9} \)
Exponent:	\(8\)\(\medspace = 2^{3} \)
Automorphism group:	$(C_2\times C_2.C_4^2).C_2^6.C_2^6.C_3.D_4.\PSL(2,7)$, of order \(1056964608\)\(\medspace = 2^{24} \cdot 3^{2} \cdot 7 \)
Composition factors:	$C_2$ x 9
Nilpotency class:	$1$
Derived length:	$1$

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

Group statistics

Order	1	2	4	8
Elements	1	31	96	384	512
Conjugacy classes	1	31	96	384	512
Divisions	1	31	48	96	176
Autjugacy classes	1	2	2	1	6

Dimension	1	2	4
Irr. complex chars.	512	0	0	512
Irr. rational chars.	32	48	96	176

Minimal presentations

Permutation degree:	$22$
Transitive degree:	$512$
Rank:	$5$
Inequivalent generating 5-tuples:	$9920$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	none	not computed	none
Arbitrary	5	not computed	not computed

Constructions

Presentation:	Abelian group $\langle a, b, c, d, e \mid a^{8}=b^{8}=c^{2}=d^{2}=e^{2}=1 \rangle$
Permutation group:	Degree $22$ $\langle(7,14,10,12,8,13,9,11), (15,22,18,20,16,21,17,19), (3,4), (5,6), (1,2), (7,10,8,9) \!\cdots\! \rangle$
Direct product:	$C_2$ ${}^3$ $\, \times\, $ $C_8$ ${}^2$
Semidirect product:	not isomorphic to a non-trivial semidirect product
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$C_2^5$ . $C_4^2$	$C_4^2$ . $C_2^5$	$(C_2^4\times C_8)$ . $C_4$	$(C_4\times C_8)$ . $C_2^4$	all 42

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

Homology

Primary decomposition:	$C_{2}^{3} \times C_{8}^{2}$
Schur multiplier:	$C_{2}^{9} \times C_{8}$
Commutator length:	$0$

Subgroups

There are 4162 subgroups, all normal (6 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_2^3\times C_8^2$	$G/Z \simeq$ $C_1$
Commutator:	$G' \simeq$ $C_1$	$G/G' \simeq$ $C_2^3\times C_8^2$
Frattini:	$\Phi \simeq$ $C_4^2$	$G/\Phi \simeq$ $C_2^5$
Fitting:	$\operatorname{Fit} \simeq$ $C_2^3\times C_8^2$	$G/\operatorname{Fit} \simeq$ $C_1$
Radical:	$R \simeq$ $C_2^3\times C_8^2$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_2^5$	$G/\operatorname{soc} \simeq$ $C_4^2$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2^3\times C_8^2$

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

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: $C_2^3\times C_8^2$

Order 256: $C_2^2\times C_8^2$ x 28, $C_2^3\times C_4\times C_8$ x 3

Order 128: $C_2\times C_8^2$ x 112, $C_2^2\times C_4\times C_8$ x 84, $C_2^4\times C_8$ x 6, $C_2^3\times C_4^2$

Order 64: $C_2\times C_4\times C_8$ x 336, $C_2^3\times C_8$ x 180, $C_8^2$ x 64, $C_2^2\times C_4^2$ x 28, $C_2^4\times C_4$ x 3

Order 32: $C_2^2\times C_8$ x 840, $C_4\times C_8$ x 192, $C_2\times C_4^2$ x 112, $C_2^3\times C_4$ x 90, $C_2^5$

Order 16: $C_2\times C_8$ x 720, $C_2^2\times C_4$ x 420, $C_4^2$ x 64, $C_2^4$ x 31

Order 8: $C_2\times C_4$ x 360, $C_2^3$ x 155, $C_8$ x 96

Order 4: $C_2^2$ x 155, $C_4$ x 48

Order 2: $C_2$ x 31

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 512: $C_2^3\times C_8^2$

Order 256: $C_2^3\times C_4\times C_8$, $C_2^2\times C_8^2$

Order 128: $C_2^2\times C_4\times C_8$ x 2, $C_2^4\times C_8$, $C_2^3\times C_4^2$, $C_2\times C_8^2$

Order 64: $C_2\times C_4\times C_8$ x 2, $C_2^3\times C_8$ x 2, $C_2^2\times C_4^2$ x 2, $C_2^4\times C_4$, $C_8^2$

Order 32: $C_2^3\times C_4$ x 3, $C_2\times C_4^2$ x 3, $C_2^2\times C_8$ x 2, $C_4\times C_8$ x 2, $C_2^5$

Order 16: $C_2^2\times C_4$ x 4, $C_4^2$ x 3, $C_2\times C_8$ x 2, $C_2^4$ x 2

Order 8: $C_2\times C_4$ x 4, $C_2^3$ x 3, $C_8$

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

Order 2: $C_2$ x 2

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 512: $C_2^3\times C_8^2$ ($C_1$)

Order 256: $C_2^2\times C_8^2$ ($C_2$) x 28, $C_2^3\times C_4\times C_8$ ($C_2$) x 3

Order 128: $C_2\times C_8^2$ ($C_2^2$) x 112, $C_2^2\times C_4\times C_8$ ($C_2^2$) x 42, $C_2^2\times C_4\times C_8$ ($C_4$) x 42, $C_2^4\times C_8$ ($C_4$) x 6, $C_2^3\times C_4^2$ ($C_2^2$)

Order 64: $C_2\times C_4\times C_8$ ($C_2\times C_4$) x 252, $C_2^3\times C_8$ ($C_8$) x 96, $C_2\times C_4\times C_8$ ($C_2^3$) x 84, $C_2^3\times C_8$ ($C_2\times C_4$) x 84, $C_8^2$ ($C_2^3$) x 64, $C_2^2\times C_4^2$ ($C_2\times C_4$) x 21, $C_2^2\times C_4^2$ ($C_2^3$) x 7, $C_2^4\times C_4$ ($C_2\times C_4$) x 3

Order 32: $C_2^2\times C_8$ ($C_2\times C_8$) x 672, $C_2^2\times C_8$ ($C_2^2\times C_4$) x 168, $C_4\times C_8$ ($C_2^2\times C_4$) x 168, $C_2\times C_4^2$ ($C_2^2\times C_4$) x 63, $C_2^3\times C_4$ ($C_2\times C_8$) x 48, $C_2\times C_4^2$ ($C_4^2$) x 42, $C_4\times C_8$ ($C_2^4$) x 24, $C_2^3\times C_4$ ($C_4^2$) x 21, $C_2^3\times C_4$ ($C_2^2\times C_4$) x 21, $C_2\times C_4^2$ ($C_2^4$) x 7, $C_2^5$ ($C_4^2$)

Order 16: $C_2\times C_8$ ($C_2^2\times C_8$) x 672, $C_2^2\times C_4$ ($C_2^2\times C_8$) x 168, $C_2^2\times C_4$ ($C_4\times C_8$) x 168, $C_2^2\times C_4$ ($C_2\times C_4^2$) x 63, $C_2\times C_8$ ($C_2^3\times C_4$) x 48, $C_4^2$ ($C_2\times C_4^2$) x 42, $C_2^4$ ($C_4\times C_8$) x 24, $C_4^2$ ($C_2^3\times C_4$) x 21, $C_2^2\times C_4$ ($C_2^3\times C_4$) x 21, $C_2^4$ ($C_2\times C_4^2$) x 7, $C_4^2$ ($C_2^5$)

Order 8: $C_2\times C_4$ ($C_2\times C_4\times C_8$) x 252, $C_8$ ($C_2^3\times C_8$) x 96, $C_2^3$ ($C_2\times C_4\times C_8$) x 84, $C_2\times C_4$ ($C_2^3\times C_8$) x 84, $C_2^3$ ($C_8^2$) x 64, $C_2\times C_4$ ($C_2^2\times C_4^2$) x 21, $C_2^3$ ($C_2^2\times C_4^2$) x 7, $C_2\times C_4$ ($C_2^4\times C_4$) x 3

Order 4: $C_2^2$ ($C_2\times C_8^2$) x 112, $C_2^2$ ($C_2^2\times C_4\times C_8$) x 42, $C_4$ ($C_2^2\times C_4\times C_8$) x 42, $C_4$ ($C_2^4\times C_8$) x 6, $C_2^2$ ($C_2^3\times C_4^2$)

Order 2: $C_2$ ($C_2^2\times C_8^2$) x 28, $C_2$ ($C_2^3\times C_4\times C_8$) x 3

Order 1: $C_1$ ($C_2^3\times C_8^2$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 512: $C_2^3\times C_8^2$ ($C_1$)

Order 256: $C_2^3\times C_4\times C_8$ ($C_2$), $C_2^2\times C_8^2$ ($C_2$)

Order 128: $C_2^4\times C_8$ ($C_4$), $C_2^3\times C_4^2$ ($C_2^2$), $C_2\times C_8^2$ ($C_2^2$), $C_2^2\times C_4\times C_8$ ($C_2^2$), $C_2^2\times C_4\times C_8$ ($C_4$)

Order 64: $C_2\times C_4\times C_8$ ($C_2^3$), $C_2\times C_4\times C_8$ ($C_2\times C_4$), $C_2^4\times C_4$ ($C_2\times C_4$), $C_2^3\times C_8$ ($C_2\times C_4$), $C_2^3\times C_8$ ($C_8$), $C_8^2$ ($C_2^3$), $C_2^2\times C_4^2$ ($C_2^3$), $C_2^2\times C_4^2$ ($C_2\times C_4$)

Order 32: $C_2^5$ ($C_4^2$), $C_2^3\times C_4$ ($C_2\times C_8$), $C_2^3\times C_4$ ($C_4^2$), $C_2^3\times C_4$ ($C_2^2\times C_4$), $C_2^2\times C_8$ ($C_2\times C_8$), $C_2^2\times C_8$ ($C_2^2\times C_4$), $C_4\times C_8$ ($C_2^4$), $C_4\times C_8$ ($C_2^2\times C_4$), $C_2\times C_4^2$ ($C_4^2$), $C_2\times C_4^2$ ($C_2^4$), $C_2\times C_4^2$ ($C_2^2\times C_4$)

Order 16: $C_2\times C_8$ ($C_2^3\times C_4$), $C_2\times C_8$ ($C_2^2\times C_8$), $C_4^2$ ($C_2^5$), $C_4^2$ ($C_2^3\times C_4$), $C_4^2$ ($C_2\times C_4^2$), $C_2^4$ ($C_4\times C_8$), $C_2^4$ ($C_2\times C_4^2$), $C_2^2\times C_4$ ($C_2^3\times C_4$), $C_2^2\times C_4$ ($C_2^2\times C_8$), $C_2^2\times C_4$ ($C_4\times C_8$), $C_2^2\times C_4$ ($C_2\times C_4^2$)

Order 8: $C_2^3$ ($C_2\times C_4\times C_8$), $C_2^3$ ($C_8^2$), $C_2^3$ ($C_2^2\times C_4^2$), $C_2\times C_4$ ($C_2\times C_4\times C_8$), $C_2\times C_4$ ($C_2^4\times C_4$), $C_2\times C_4$ ($C_2^3\times C_8$), $C_2\times C_4$ ($C_2^2\times C_4^2$), $C_8$ ($C_2^3\times C_8$)

Order 4: $C_2^2$ ($C_2^3\times C_4^2$), $C_2^2$ ($C_2\times C_8^2$), $C_2^2$ ($C_2^2\times C_4\times C_8$), $C_4$ ($C_2^4\times C_8$), $C_4$ ($C_2^2\times C_4\times C_8$)

Order 2: $C_2$ ($C_2^3\times C_4\times C_8$), $C_2$ ($C_2^2\times C_8^2$)

Order 1: $C_1$ ($C_2^3\times C_8^2$)

Series

Derived series

$C_2^3\times C_8^2$

$\rhd$

$C_1$

Chief series

$C_2^3\times C_8^2$

$\rhd$

$C_2^3\times C_4\times C_8$

$\rhd$

$C_2^3\times C_4^2$

$\rhd$

$C_2^2\times C_4^2$

$\rhd$

$C_2\times C_4^2$

$\rhd$

$C_4^2$

$\rhd$

$C_2\times C_4$

$\rhd$

$C_2^2$

$\rhd$

$C_2$

$\rhd$

$C_1$

Lower central series

$C_2^3\times C_8^2$

$\rhd$

$C_1$

Upper central series

$C_1$

$\lhd$

$C_2^3\times C_8^2$

Supergroups

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

This group is a maximal quotient of 1 larger groups in the database.

Character theory

Complex character table

The $512 \times 512$ character table is not available for this group.

Rational character table

See the $176 \times 176$ rational character table (warning: may be slow to load).