Abstract group 512.6276915: $C_2^2\times C_4^2\times C_8$

• Label: 512.6276915
• Order: \( 2^{9} \)
• Exponent: \( 2^{3} \)
• Abelian: yes
• $\card{\Aut(G)}$: \( 2^{28} \cdot 3^{2} \)
• Perm deg.: $20$
• Trans deg.: $512$
• Rank: $5$

Group information

Description:	$C_2^2\times C_4^2\times C_8$
Order:	\(512\)\(\medspace = 2^{9} \)
Exponent:	\(8\)\(\medspace = 2^{3} \)
Automorphism group:	$C_2^7.C_2^6.C_2.C_2^6.C_2^6.S_3^2$, of order \(2415919104\)\(\medspace = 2^{28} \cdot 3^{2} \)
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	224	256	512
Conjugacy classes	1	31	224	256	512
Divisions	1	31	112	64	208
Autjugacy classes	1	3	3	1	8

Dimension	1	2	4
Irr. complex chars.	512	0	0	512
Irr. rational chars.	32	112	64	208

Minimal presentations

Permutation degree:	$20$
Transitive degree:	$512$
Rank:	$5$
Inequivalent generating 5-tuples:	$4340$

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 $\langle a, b, c, d, e \mid a^{8}=b^{4}=c^{4}=d^{2}=e^{2}=1 \rangle$
Permutation group:	Degree $20$ $\langle(13,20,16,18,14,19,15,17), (5,8,6,7), (9,12,10,11), (3,4), (1,2), (13,16,14,15)(17,20,18,19), (5,6)(7,8), (9,10)(11,12), (13,14)(15,16)(17,18)(19,20)\rangle$
Direct product:	$C_2$ ${}^2$ $\, \times\, $ $C_4$ ${}^2$ $\, \times\, $ $C_8$
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:	$C_4^3$ . $C_2^3$	$C_2^3$ . $C_4^3$	$(C_2\times C_4^3)$ . $C_4$	$C_4^3$ . $(C_2\times C_4)$	all 54

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

Homology

Primary decomposition:	$C_{2}^{2} \times C_{4}^{2} \times C_{8}$
Schur multiplier:	$C_{2}^{7} \times C_{4}^{3}$
Commutator length:	$0$

Subgroups

There are 6594 subgroups, all normal (8 characteristic).

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

Special subgroups

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

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: $C_2^2\times C_4^2\times C_8$

Order 256: $C_2\times C_4^2\times C_8$ x 24, $C_2^3\times C_4\times C_8$ x 6, $C_2^2\times C_4^3$

Order 128: $C_2^2\times C_4\times C_8$ x 168, $C_4^2\times C_8$ x 64, $C_2\times C_4^3$ x 24, $C_2^3\times C_4^2$ x 7, $C_2^4\times C_8$ x 4

Order 64: $C_2\times C_4\times C_8$ x 672, $C_2^2\times C_4^2$ x 196, $C_2^3\times C_8$ x 120, $C_4^3$ x 64, $C_2^4\times C_4$ x 7

Order 32: $C_2\times C_4^2$ x 784, $C_2^2\times C_8$ x 560, $C_4\times C_8$ x 384, $C_2^3\times C_4$ x 210, $C_2^5$

Order 16: $C_2^2\times C_4$ x 980, $C_2\times C_8$ x 480, $C_4^2$ x 448, $C_2^4$ x 31

Order 8: $C_2\times C_4$ x 840, $C_2^3$ x 155, $C_8$ x 64

Order 4: $C_2^2$ x 155, $C_4$ x 112

Order 2: $C_2$ x 31

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 512: $C_2^2\times C_4^2\times C_8$

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

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

Order 64: $C_2^2\times C_4^2$ x 5, $C_2\times C_4\times C_8$ x 2, $C_4^3$ x 2, $C_2^4\times C_4$ x 2, $C_2^3\times C_8$ x 2

Order 32: $C_2^3\times C_4$ x 6, $C_2\times C_4^2$ x 6, $C_2^2\times C_8$ x 3, $C_2^5$, $C_4\times C_8$

Order 16: $C_2^2\times C_4$ x 9, $C_4^2$ x 3, $C_2^4$ x 3, $C_2\times C_8$ x 2

Order 8: $C_2\times C_4$ x 7, $C_2^3$ x 5, $C_8$

Order 4: $C_2^2$ x 5, $C_4$ x 3

Order 2: $C_2$ x 3

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 512: $C_2^2\times C_4^2\times C_8$ ($C_1$)

Order 256: $C_2\times C_4^2\times C_8$ ($C_2$) x 24, $C_2^3\times C_4\times C_8$ ($C_2$) x 6, $C_2^2\times C_4^3$ ($C_2$)

Order 128: $C_2^2\times C_4\times C_8$ ($C_4$) x 96, $C_2^2\times C_4\times C_8$ ($C_2^2$) x 72, $C_4^2\times C_8$ ($C_2^2$) x 64, $C_2\times C_4^3$ ($C_2^2$) x 12, $C_2\times C_4^3$ ($C_4$) x 12, $C_2^4\times C_8$ ($C_2^2$) x 4, $C_2^3\times C_4^2$ ($C_4$) x 4, $C_2^3\times C_4^2$ ($C_2^2$) x 3

Order 64: $C_2\times C_4\times C_8$ ($C_2\times C_4$) x 576, $C_2^2\times C_4^2$ ($C_2\times C_4$) x 114, $C_2\times C_4\times C_8$ ($C_2^3$) x 96, $C_2^3\times C_8$ ($C_2\times C_4$) x 96, $C_2^2\times C_4^2$ ($C_8$) x 64, $C_4^3$ ($C_2\times C_4$) x 48, $C_2^3\times C_8$ ($C_2^3$) x 24, $C_2^2\times C_4^2$ ($C_2^3$) x 18, $C_4^3$ ($C_2^3$) x 16, $C_2^4\times C_4$ ($C_2\times C_4$) x 6, $C_2^4\times C_4$ ($C_2^3$)

Order 32: $C_4\times C_8$ ($C_2^2\times C_4$) x 384, $C_2\times C_4^2$ ($C_2\times C_8$) x 384, $C_2^2\times C_8$ ($C_2^2\times C_4$) x 288, $C_2^2\times C_8$ ($C_4^2$) x 256, $C_2\times C_4^2$ ($C_2^2\times C_4$) x 244, $C_2\times C_4^2$ ($C_4^2$) x 144, $C_2^3\times C_4$ ($C_2\times C_8$) x 96, $C_2^3\times C_4$ ($C_2^2\times C_4$) x 63, $C_2^3\times C_4$ ($C_4^2$) x 48, $C_2^2\times C_8$ ($C_2^4$) x 16, $C_2\times C_4^2$ ($C_2^4$) x 12, $C_2^3\times C_4$ ($C_2^4$) x 3, $C_2^5$ ($C_2^2\times C_4$)

Order 16: $C_2\times C_8$ ($C_2\times C_4^2$) x 384, $C_2^2\times C_4$ ($C_4\times C_8$) x 384, $C_2^2\times C_4$ ($C_2^2\times C_8$) x 288, $C_4^2$ ($C_2^2\times C_8$) x 256, $C_2^2\times C_4$ ($C_2\times C_4^2$) x 244, $C_4^2$ ($C_2\times C_4^2$) x 144, $C_2\times C_8$ ($C_2^3\times C_4$) x 96, $C_2^2\times C_4$ ($C_2^3\times C_4$) x 63, $C_4^2$ ($C_2^3\times C_4$) x 48, $C_2^4$ ($C_2^2\times C_8$) x 16, $C_2^4$ ($C_2\times C_4^2$) x 12, $C_2^4$ ($C_2^3\times C_4$) x 3, $C_2^2\times C_4$ ($C_2^5$)

Order 8: $C_2\times C_4$ ($C_2\times C_4\times C_8$) x 576, $C_2\times C_4$ ($C_2^2\times C_4^2$) x 114, $C_2^3$ ($C_2\times C_4\times C_8$) x 96, $C_2\times C_4$ ($C_2^3\times C_8$) x 96, $C_8$ ($C_2^2\times C_4^2$) x 64, $C_2\times C_4$ ($C_4^3$) x 48, $C_2^3$ ($C_2^3\times C_8$) x 24, $C_2^3$ ($C_2^2\times C_4^2$) x 18, $C_2^3$ ($C_4^3$) x 16, $C_2\times C_4$ ($C_2^4\times C_4$) x 6, $C_2^3$ ($C_2^4\times C_4$)

Order 4: $C_4$ ($C_2^2\times C_4\times C_8$) x 96, $C_2^2$ ($C_2^2\times C_4\times C_8$) x 72, $C_2^2$ ($C_4^2\times C_8$) x 64, $C_2^2$ ($C_2\times C_4^3$) x 12, $C_4$ ($C_2\times C_4^3$) x 12, $C_2^2$ ($C_2^4\times C_8$) x 4, $C_4$ ($C_2^3\times C_4^2$) x 4, $C_2^2$ ($C_2^3\times C_4^2$) x 3

Order 2: $C_2$ ($C_2\times C_4^2\times C_8$) x 24, $C_2$ ($C_2^3\times C_4\times C_8$) x 6, $C_2$ ($C_2^2\times C_4^3$)

Order 1: $C_1$ ($C_2^2\times C_4^2\times C_8$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 512: $C_2^2\times C_4^2\times C_8$ ($C_1$)

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

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

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

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

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

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

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

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

Order 1: $C_1$ ($C_2^2\times C_4^2\times C_8$)

Series

Derived series

$C_2^2\times C_4^2\times C_8$

$\rhd$

$C_1$

Chief series

$C_2^2\times C_4^2\times C_8$

$\rhd$

$C_2^2\times C_4^3$

$\rhd$

$C_2^3\times C_4^2$

$\rhd$

$C_2^4\times C_4$

$\rhd$

$C_2^3\times C_4$

$\rhd$

$C_2^2\times C_4$

$\rhd$

$C_2^3$

$\rhd$

$C_2^2$

$\rhd$

$C_2$

$\rhd$

$C_1$

Lower central series

$C_2^2\times C_4^2\times C_8$

$\rhd$

$C_1$

Upper central series

$C_1$

$\lhd$

$C_2^2\times C_4^2\times C_8$

Supergroups

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

This group is a maximal quotient of 0 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 $208 \times 208$ rational character table (warning: may be slow to load).