Abstract group 512.51562: $C_2^6:D_4$

• Label: 512.51562
• Order: \( 2^{9} \)
• Exponent: \( 2^{3} \)
• Nilpotent: yes
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{3} \)
• $\card{Z(G)}$: \( 2^{2} \)
• $\card{\Aut(G)}$: \( 2^{15} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{8} \)
• 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:	$C_2^9.D_4^2$, of order \(32768\)\(\medspace = 2^{15} \)
Composition factors:	$C_2$ x 9
Nilpotency class:	$4$
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	127	320	64	512
Conjugacy classes	1	24	17	2	44
Divisions	1	24	17	1	43
Autjugacy classes	1	9	7	1	18

Dimension	1	2	4	8
Irr. complex chars.	8	14	20	2	44
Irr. rational chars.	8	12	21	2	43

Minimal presentations

Permutation degree:	$16$
Transitive degree:	$16$
Rank:	$3$
Inequivalent generating triples:	$1344$

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:	${\langle a, b, c, d, e, f \mid a^{2}=b^{4}=c^{4}=d^{2}=e^{2}=f^{4}=[a,d]= \!\cdots\! \rangle}$
Permutation group:	Degree $16$ $\langle(1,9)(2,10)(3,11)(4,12)(5,13)(6,14)(7,15)(8,16), (1,4)(2,3)(5,8)(6,7)(9,16) \!\cdots\! \rangle$
Transitive group:	16T969				more information
Direct product:	not isomorphic to a non-trivial direct product
Semidirect product:	$C_2^6$ $\,\rtimes\,$ $D_4$	$(D_4:C_2^3)$ $\,\rtimes\,$ $D_4$	$(C_2^5:D_4)$ $\,\rtimes\,$ $C_2$	$(C_2^5.D_4)$ $\,\rtimes\,$ $C_2$	all 11
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$C_2^5$ . $(C_2\times D_4)$	$C_2^4$ . $(C_4:D_4)$	$C_2$ . $(C_2^5:D_4)$	$C_2$ . $(C_2^5:D_4)$	all 21
Aut. group:	$\Aut(C_2^3:C_4)$

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

Homology

Abelianization:	$C_{2}^{3} $
Schur multiplier:	$C_{2}^{6}$
Commutator length:	$1$

Subgroups

There are 11414 subgroups in 2083 conjugacy classes, 60 normal (20 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_2^2$	$G/Z \simeq$ $C_2^4:D_4$
Commutator:	$G' \simeq$ $D_4\times C_2^3$	$G/G' \simeq$ $C_2^3$
Frattini:	$\Phi \simeq$ $D_4\times C_2^3$	$G/\Phi \simeq$ $C_2^3$
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:	$\operatorname{soc} \simeq$ $C_2^2$	$G/\operatorname{soc} \simeq$ $C_2^4:D_4$
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

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Normal subgroups up to automorphism

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Classes of subgroups up to conjugation

Order 512: $C_2^6:D_4$

Order 256: $C_2^6:C_4$ x 2, $C_2^5:D_4$ x 2, $C_2^6:C_2^2$, $C_2^5.D_4$, $C_2^5.D_4$

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

Order 64: $C_2^3:D_4$ x 50, $C_2^4:C_4$ x 10, $C_2^4:C_4$ x 10, $C_2^4:C_4$ x 10, $C_2\wr C_4$ x 8, $C_2\wr C_2^2$ x 8, $D_4\times C_2^3$ x 5, $C_2^2.C_4^2$ x 4, $C_2^2.C_4^2$ x 4, $\OD_{16}:C_2^2$ x 2, $C_2^2.Q_{16}$ x 2, $C_2^2.D_8$ x 2, $C_2^6$ x 2, $D_4:D_4$ x 2, $D_4^2$ x 2, $C_2^3:D_4$ x 2, $C_2^2:\OD_{16}$, $D_4:C_2^3$, $C_4^2:C_2^2$, $C_4^2:C_2^2$, $C_4^2:C_2^2$

Order 32: $C_2^2\wr C_2$ x 141, $C_2^3:C_4$ x 82, $C_2^2\times D_4$ x 77, $C_2^5$ x 32, $C_2^3:C_4$ x 16, $C_4:D_4$ x 8, $C_2^3\times C_4$ x 5, $\OD_{16}:C_2$ x 4, $D_4:C_2^2$ x 4, $C_4\times D_4$ x 4, $C_2.C_4^2$ x 4, $C_2^2.D_4$ x 3, $C_2^2:C_8$ x 2, $D_4:C_2^2$ x 2, $C_2^2:Q_8$ x 2, $C_2\times \OD_{16}$, $C_4:D_4$, $C_4^2:C_2$, $C_4^2:C_2$

Order 16: $C_2^4$ x 229, $C_2\times D_4$ x 222, $C_2^2:C_4$ x 127, $C_2^2\times C_4$ x 49, $D_4:C_2$ x 6, $C_4:C_4$ x 4, $\OD_{16}$ x 2, $C_2\times C_8$, $C_4^2$, $C_2\times Q_8$

Order 8: $C_2^3$ x 448, $D_4$ x 105, $C_2\times C_4$ x 73, $Q_8$, $C_8$

Order 4: $C_2^2$ x 214, $C_4$ x 17

Order 2: $C_2$ x 24

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 512: $C_2^6:D_4$

Order 256: $C_2^6:C_2^2$, $C_2^6:C_4$, $C_2^5.D_4$, $C_2^5:D_4$, $C_2^5.D_4$

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

Order 64: $C_2^3:D_4$ x 10, $C_2^4:C_4$ x 4, $C_2^4:C_4$ x 3, $C_2^4:C_4$ x 3, $D_4\times C_2^3$ x 3, $C_2^2.C_4^2$ x 2, $\OD_{16}:C_2^2$, $C_2^2.Q_{16}$, $C_2^2:\OD_{16}$, $C_2^2.D_8$, $C_2\wr C_4$, $C_2^6$, $D_4:C_2^3$, $C_2^2.C_4^2$, $D_4:D_4$, $D_4^2$, $C_4^2:C_2^2$, $C_2^3:D_4$, $C_4^2:C_2^2$, $C_4^2:C_2^2$, $C_2\wr C_2^2$

Order 32: $C_2^2\times D_4$ x 20, $C_2^3:C_4$ x 18, $C_2^2\wr C_2$ x 17, $C_2^5$ x 9, $C_2^3:C_4$ x 3, $C_2^3\times C_4$ x 3, $C_4:D_4$ x 3, $D_4:C_2^2$ x 2, $C_2^2.D_4$ x 2, $C_4\times D_4$ x 2, $C_2.C_4^2$ x 2, $\OD_{16}:C_2$, $C_2^2:C_8$, $D_4:C_2^2$, $C_2\times \OD_{16}$, $C_4:D_4$, $C_4^2:C_2$, $C_2^2:Q_8$, $C_4^2:C_2$

Order 16: $C_2^4$ x 40, $C_2\times D_4$ x 32, $C_2^2:C_4$ x 19, $C_2^2\times C_4$ x 18, $C_4:C_4$ x 2, $D_4:C_2$ x 2, $\OD_{16}$, $C_2\times C_8$, $C_4^2$, $C_2\times Q_8$

Order 8: $C_2^3$ x 65, $C_2\times C_4$ x 22, $D_4$ x 13, $Q_8$, $C_8$

Order 4: $C_2^2$ x 38, $C_4$ x 7

Order 2: $C_2$ x 9

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

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

Order 256: $C_2^6:C_4$ ($C_2$) x 2, $C_2^5:D_4$ ($C_2$) x 2, $C_2^6:C_2^2$ ($C_2$), $C_2^5.D_4$ ($C_2$), $C_2^5.D_4$ ($C_2$)

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

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

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

Order 16: $C_2^4$ ($C_2^2\wr C_2$) x 3, $C_2^4$ ($C_4:D_4$) x 2, $C_2^2\times C_4$ ($C_4:D_4$), $C_2^2\times C_4$ ($C_4:D_4$)

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

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

Order 2: $C_2$ ($C_2^5:D_4$) x 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: $C_2^6:C_2^2$ ($C_2$), $C_2^6:C_4$ ($C_2$), $C_2^5.D_4$ ($C_2$), $C_2^5:D_4$ ($C_2$), $C_2^5.D_4$ ($C_2$)

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

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

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

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

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

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

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

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

Series

Derived series

$C_2^6:D_4$

$\rhd$

$D_4\times C_2^3$

$\rhd$

$C_2$

$\rhd$

$C_1$

Chief series

$C_2^6:D_4$

$\rhd$

$C_2^6:C_2^2$

$\rhd$

$C_2^4:D_4$

$\rhd$

$D_4\times C_2^3$

$\rhd$

$C_2^2\times D_4$

$\rhd$

$C_2^4$

$\rhd$

$C_2^3$

$\rhd$

$C_2^2$

$\rhd$

$C_2$

$\rhd$

$C_1$

Lower central series

$C_2^6:D_4$

$\rhd$

$D_4\times C_2^3$

$\rhd$

$C_2^3$

$\rhd$

$C_2$

$\rhd$

$C_1$

Upper central series

$C_1$

$\lhd$

$C_2^2$

$\lhd$

$C_2^4$

$\lhd$

$D_4\times C_2^3$

$\lhd$

$C_2^6:D_4$

Supergroups

This group is a maximal subgroup of 6 larger groups in the database.

This group is a maximal quotient of 11 larger groups in the database.

Character theory

Complex character table

See the $44 \times 44$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $43 \times 43$ rational character table.