Abstract group 512.57589: $C_2^6.D_4$

• Label: 512.57589
• Order: \( 2^{9} \)
• Exponent: \( 2^{3} \)
• Nilpotent: yes
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{4} \)
• $\card{Z(G)}$: \( 2 \)
• $\card{\Aut(G)}$: \( 2^{12} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{4} \)
• Perm deg.: $16$
• Trans deg.: $16$
• Rank: $3$

Group information

Description:	$C_2^6.D_4$
Order:	\(512\)\(\medspace = 2^{9} \)
Exponent:	\(8\)\(\medspace = 2^{3} \)
Automorphism group:	$C_2^3.D_4^2:D_4$, of order \(4096\)\(\medspace = 2^{12} \)
Composition factors:	$C_2$ x 9
Nilpotency class:	$5$
Derived length:	$3$

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

Group statistics

Order	1	2	4	8
Elements	1	79	304	128	512
Conjugacy classes	1	17	19	4	41
Divisions	1	17	13	2	33
Autjugacy classes	1	13	9	1	24

Dimension	1	2	4	8
Irr. complex chars.	16	12	8	5	41
Irr. rational chars.	8	8	12	5	33

Minimal presentations

Permutation degree:	$16$
Transitive degree:	$16$
Rank:	$3$
Inequivalent generating triples:	$10752$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	8	8	8
Arbitrary	8	8	8

Constructions

Presentation:	${\langle a, b, c, d, e, f \mid a^{4}=b^{4}=c^{2}=d^{2}=e^{2}=f^{4}=[a,c]= \!\cdots\! \rangle}$
Permutation group:	Degree $16$ $\langle(1,9)(2,10)(3,11)(4,12)(5,15,7,13)(6,16,8,14), (1,5)(2,6)(3,7)(4,8)(9,13) \!\cdots\! \rangle$
Transitive group:	16T834	16T862	16T897	16T900	all 5
Direct product:	not isomorphic to a non-trivial direct product
Semidirect product:	$(C_2^4:Q_8)$ $\,\rtimes\,$ $C_4$	$(C_2^4.D_8)$ $\,\rtimes\,$ $C_2$ (2)	$(C_2^4:D_4)$ $\,\rtimes\,$ $C_4$	$(C_2^4.D_4)$ $\,\rtimes\,$ $C_4$	all 11
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$C_2^6$ . $D_4$	$(C_2^4:C_4)$ . $D_4$ (2)	$(C_2^4:C_4)$ . $D_4$	$C_2^5$ . $(C_2\times D_4)$	all 30

Elements of the group are displayed as permutations of degree 16.

Homology

Abelianization:	$C_{2}^{2} \times C_{4} $
Schur multiplier:	$C_{2}^{4}$
Commutator length:	$1$

Subgroups

There are 5474 subgroups in 920 conjugacy classes, 54 normal (44 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_2$	$G/Z \simeq$ $C_2^5.D_4$
Commutator:	$G' \simeq$ $C_2^3:C_4$	$G/G' \simeq$ $C_2^2\times C_4$
Frattini:	$\Phi \simeq$ $C_2^4:C_4$	$G/\Phi \simeq$ $C_2^3$
Fitting:	$\operatorname{Fit} \simeq$ $C_2^6.D_4$	$G/\operatorname{Fit} \simeq$ $C_1$
Radical:	$R \simeq$ $C_2^6.D_4$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_2$	$G/\operatorname{soc} \simeq$ $C_2^5.D_4$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2^6.D_4$

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

Normal subgroups up to automorphism

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Classes of subgroups up to conjugation

Order 512: $C_2^6.D_4$

Order 256: $C_2^4.Q_{16}$ x 2, $C_2^4.D_8$ x 2, $C_2^4:\OD_{16}$, $C_2^4.C_2^4$, $C_2^4.C_4^2$

Order 128: $C_2^4:C_8$ x 3, $C_2^3.C_4^2$ x 3, $C_2^4:Q_8$ x 2, $C_2^4:D_4$ x 2, $C_2^4.D_4$ x 2, $C_2^5:C_4$ x 2, $C_2^4.D_4$, $C_2^4.D_4$, $C_4^2:C_2^3$, $C_2^3:\OD_{16}$, $C_2^3\wr C_2$

Order 64: $C_2^4:C_4$ x 16, $C_2^4:C_4$ x 7, $C_2^2.C_4^2$ x 5, $C_2^3:D_4$ x 5, $C_2^3:C_8$ x 3, $C_2^4:C_4$ x 2, $C_4^2:C_2^2$ x 2, $D_4:D_4$ x 2, $C_2^2:\OD_{16}$, $C_2^6$, $C_2^3:Q_8$, $C_4^2:C_2^2$, $C_2^3:D_4$, $C_4^2:C_2^2$, $C_4^2:C_2^2$

Order 32: $C_2^3:C_4$ x 57, $C_2^2\wr C_2$ x 19, $C_2^5$ x 16, $C_4\times D_4$ x 6, $C_2^2\times D_4$ x 5, $C_2^2.D_4$ x 5, $C_2^3\times C_4$ x 4, $C_2^2:Q_8$ x 4, $C_4:D_4$ x 4, $C_2^2:C_8$ x 3, $C_2^3:C_4$ x 2, $C_4^2:C_2$ x 2, $C_4^2:C_2$ x 2, $C_4^2:C_2$ x 2, $D_4:C_2^2$, $C_2\times \OD_{16}$

Order 16: $C_2^4$ x 115, $C_2^2:C_4$ x 69, $C_2^2\times C_4$ x 35, $C_2\times D_4$ x 27, $C_4:C_4$ x 6, $\OD_{16}$ x 2, $C_2\times C_8$ x 2, $C_4^2$ x 2, $D_4:C_2$ x 2, $C_2\times Q_8$

Order 8: $C_2^3$ x 229, $C_2\times C_4$ x 49, $D_4$ x 19, $C_8$ x 2, $Q_8$

Order 4: $C_2^2$ x 119, $C_4$ x 13

Order 2: $C_2$ x 17

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 512: $C_2^6.D_4$

Order 256: $C_2^4:\OD_{16}$, $C_2^4.Q_{16}$, $C_2^4.D_8$, $C_2^4.C_2^4$, $C_2^4.C_4^2$

Order 128: $C_2^4:Q_8$ x 2, $C_2^4:D_4$ x 2, $C_2^4.D_4$ x 2, $C_2^4:C_8$ x 2, $C_2^3.C_4^2$ x 2, $C_2^5:C_4$ x 2, $C_2^4.D_4$, $C_2^4.D_4$, $C_4^2:C_2^3$, $C_2^3:\OD_{16}$, $C_2^3\wr C_2$

Order 64: $C_2^4:C_4$ x 13, $C_2^4:C_4$ x 6, $C_2^2.C_4^2$ x 5, $C_2^3:D_4$ x 4, $C_2^4:C_4$ x 2, $C_2^3:C_8$ x 2, $C_2^2:\OD_{16}$, $C_2^6$, $C_4^2:C_2^2$, $D_4:D_4$, $C_2^3:Q_8$, $C_4^2:C_2^2$, $C_2^3:D_4$, $C_4^2:C_2^2$, $C_4^2:C_2^2$

Order 32: $C_2^3:C_4$ x 33, $C_2^5$ x 12, $C_2^2\wr C_2$ x 11, $C_2^2\times D_4$ x 4, $C_2^3\times C_4$ x 3, $C_2^2.D_4$ x 3, $C_2^2:Q_8$ x 3, $C_4:D_4$ x 3, $C_4\times D_4$ x 3, $C_2^3:C_4$ x 2, $C_2^2:C_8$ x 2, $C_4^2:C_2$ x 2, $D_4:C_2^2$, $C_2\times \OD_{16}$, $C_4^2:C_2$, $C_4^2:C_2$

Order 16: $C_2^4$ x 59, $C_2^2:C_4$ x 34, $C_2^2\times C_4$ x 19, $C_2\times D_4$ x 16, $C_4:C_4$ x 4, $\OD_{16}$, $C_2\times C_8$, $C_4^2$, $D_4:C_2$, $C_2\times Q_8$

Order 8: $C_2^3$ x 100, $C_2\times C_4$ x 26, $D_4$ x 11, $Q_8$, $C_8$

Order 4: $C_2^2$ x 62, $C_4$ x 9

Order 2: $C_2$ x 13

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 512: $C_2^6.D_4$ ($C_1$)

Order 256: $C_2^4.Q_{16}$ ($C_2$) x 2, $C_2^4.D_8$ ($C_2$) x 2, $C_2^4:\OD_{16}$ ($C_2$), $C_2^4.C_2^4$ ($C_2$), $C_2^4.C_4^2$ ($C_2$)

Order 128: $C_2^4:C_8$ ($C_2^2$) x 2, $C_2^3.C_4^2$ ($C_2^2$) x 2, $C_2^4:Q_8$ ($C_2^2$), $C_2^4:Q_8$ ($C_4$), $C_2^4:D_4$ ($C_2^2$), $C_2^4:D_4$ ($C_4$), $C_2^4.D_4$ ($C_4$), $C_4^2:C_2^3$ ($C_4$), $C_2^5:C_4$ ($C_2^2$)

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

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

Order 16: $C_2^4$ ($C_2^3:C_4$) x 2, $C_2^4$ ($C_4\wr C_2$) x 2, $C_2^4$ ($Q_{16}:C_2$), $C_2^4$ ($D_8:C_2$), $C_2^4$ ($C_2^3:C_4$)

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

Order 4: $C_2^2$ ($C_2^4.D_4$), $C_2^2$ ($C_2^5:C_4$), $C_2^2$ ($C_2^4.D_4$)

Order 2: $C_2$ ($C_2^5.D_4$)

Order 1: $C_1$ ($C_2^6.D_4$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 512: $C_2^6.D_4$ ($C_1$)

Order 256: $C_2^4:\OD_{16}$ ($C_2$), $C_2^4.Q_{16}$ ($C_2$), $C_2^4.D_8$ ($C_2$), $C_2^4.C_2^4$ ($C_2$), $C_2^4.C_4^2$ ($C_2$)

Order 128: $C_2^4:Q_8$ ($C_2^2$), $C_2^4:Q_8$ ($C_4$), $C_2^4:D_4$ ($C_2^2$), $C_2^4:D_4$ ($C_4$), $C_2^4.D_4$ ($C_4$), $C_2^4:C_8$ ($C_2^2$), $C_2^3.C_4^2$ ($C_2^2$), $C_4^2:C_2^3$ ($C_4$), $C_2^5:C_4$ ($C_2^2$)

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

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

Order 16: $C_2^4$ ($C_2^3:C_4$) x 2, $C_2^4$ ($C_4\wr C_2$) x 2, $C_2^4$ ($Q_{16}:C_2$), $C_2^4$ ($D_8:C_2$), $C_2^4$ ($C_2^3:C_4$)

Order 8: $C_2^3$ ($C_2^3.D_4$), $C_2^3$ ($C_2^4:C_4$), $C_2^3$ ($C_2\wr C_4$), $C_2^3$ ($C_4^2:C_2^2$)

Order 4: $C_2^2$ ($C_2^4.D_4$), $C_2^2$ ($C_2^5:C_4$), $C_2^2$ ($C_2^4.D_4$)

Order 2: $C_2$ ($C_2^5.D_4$)

Order 1: $C_1$ ($C_2^6.D_4$)

Series

Derived series

$C_2^6.D_4$

$\rhd$

$C_2^3:C_4$

$\rhd$

$C_2$

$\rhd$

$C_1$

Chief series

$C_2^6.D_4$

$\rhd$

$C_2^4.C_4^2$

$\rhd$

$C_2^5:C_4$

$\rhd$

$C_2^4:C_4$

$\rhd$

$C_2^3:C_4$

$\rhd$

$C_2^4$

$\rhd$

$C_2^3$

$\rhd$

$C_2^2$

$\rhd$

$C_2$

$\rhd$

$C_1$

Lower central series

$C_2^6.D_4$

$\rhd$

$C_2^3:C_4$

$\rhd$

$C_2^4$

$\rhd$

$C_2^2$

$\rhd$

$C_2$

$\rhd$

$C_1$

Upper central series

$C_1$

$\lhd$

$C_2$

$\lhd$

$C_2^3$

$\lhd$

$C_2^5$

$\lhd$

$C_2^5:C_4$

$\lhd$

$C_2^6.D_4$

Supergroups

This group is a maximal subgroup of 5 larger groups in the database.

This group is a maximal quotient of 7 larger groups in the database.

Character theory

Complex character table

See the $41 \times 41$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $33 \times 33$ rational character table.