Abstract group 512.6338937: $C_2^6.D_4$

• Label: 512.6338937
• Order: \( 2^{9} \)
• Exponent: \( 2^{3} \)
• Nilpotent: yes
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{5} \)
• $\card{Z(G)}$: \( 2^{4} \)
• $\card{\Aut(G)}$: \( 2^{19} \cdot 3 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{14} \cdot 3 \)
• Perm deg.: $20$
• Trans deg.: $64$
• Rank: $5$

Group information

Description:	$C_2^6.D_4$
Order:	\(512\)\(\medspace = 2^{9} \)
Exponent:	\(8\)\(\medspace = 2^{3} \)
Automorphism group:	$C_2^5.C_2^6.C_6.C_2^6.C_2$, of order \(1572864\)\(\medspace = 2^{19} \cdot 3 \)
Composition factors:	$C_2$ x 9
Nilpotency class:	$3$
Derived length:	$2$

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

Group statistics

Order	1	2	4	8
Elements	1	127	256	128	512
Conjugacy classes	1	35	52	16	104
Divisions	1	35	44	12	92
Autjugacy classes	1	8	9	2	20

Dimension	1	2	4	8
Irr. complex chars.	32	56	16	0	104
Irr. rational chars.	32	48	4	8	92

Minimal presentations

Permutation degree:	$20$
Transitive degree:	$64$
Rank:	$5$
Inequivalent generating 5-tuples:	$6666240$

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:	${\langle a, b, c, d, e, f \mid a^{2}=b^{4}=c^{8}=d^{2}=e^{2}=f^{2}=[a,d]= \!\cdots\! \rangle}$
Permutation group:	Degree $20$ $\langle(1,2)(3,9)(4,8)(5,13)(6,10)(7,12)(11,16)(14,15), (1,3)(4,5)(6,14)(7,10) \!\cdots\! \rangle$
Direct product:	$C_2$ ${}^2$ $\, \times\, $ $(C_4^2:D_4)$
Semidirect product:	$(Q_8\times C_2^3)$ $\,\rtimes\,$ $D_4$	$(D_4:C_2^3)$ $\,\rtimes\,$ $D_4$	$(D_4\times C_2^3)$ $\,\rtimes\,$ $D_4$	$(D_8:C_2^4)$ $\,\rtimes\,$ $C_2$	all 33
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$C_2^6$ . $D_4$	$(C_2^4\times C_4)$ . $D_4$ (2)	$C_2^5$ . $(C_2\times D_4)$ (2)	$C_2^4$ . $(C_4:D_4)$ (2)	all 44
Aut. group:	$\Aut(C_4\times C_{48})$	$\Aut(C_{16}:C_{12})$

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

Homology

Abelianization:	$C_{2}^{5} $
Schur multiplier:	$C_{2}^{11}$
Commutator length:	$1$

Subgroups

There are 20402 subgroups in 8134 conjugacy classes, 994 normal (26 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_2^2\times C_4$	$G/Z \simeq$ $C_4:D_4$
Commutator:	$G' \simeq$ $C_2^2\times C_4$	$G/G' \simeq$ $C_2^5$
Frattini:	$\Phi \simeq$ $C_2^2\times C_4$	$G/\Phi \simeq$ $C_2^5$
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:	$\operatorname{soc} \simeq$ $C_2^3$	$G/\operatorname{soc} \simeq$ $C_2^3:D_4$
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: $C_2^5.D_4$ x 24, $C_4^2:C_2^4$ x 2, $C_4^2:C_2^4$ x 2, $D_8:C_2^4$, $C_2^4:\OD_{16}$, $C_2^5.D_4$

Order 128: $C_4^2:D_4$ x 64, $C_4^2:C_2^3$ x 60, $C_4^2:C_2^3$ x 36, $D_8:C_2^3$ x 30, $C_2^3:\OD_{16}$ x 18, $C_2^4.D_4$ x 12, $C_4^2:C_2^3$ x 4, $C_4^2:C_2^3$ x 4, $D_4:C_2^4$ x 2, $Q_{16}:C_2^3$ x 2, $C_8:C_2^4$ x 2, $D_8:C_2^3$ x 2, $C_2^3\times \OD_{16}$ x 2, $C_2^4.D_4$ x 2, $C_2^4:Q_8$ x 2, $C_2^4:D_4$ x 2, $C_2^4:D_4$ x 2, $C_4^2:C_2^3$ x 2, $C_2^4:C_8$ x 2, $C_2^5\times C_4$

Order 64: $C_4^2:C_2^2$ x 272, $D_8:C_2^2$ x 136, $C_4^2:C_2^2$ x 96, $C_4^2:C_2^2$ x 80, $D_4:C_2^3$ x 68, $Q_{16}:C_2^2$ x 48, $C_8:C_2^3$ x 48, $D_8:C_2^2$ x 48, $C_2^3.D_4$ x 48, $C_2^3:Q_8$ x 48, $C_2^3:D_4$ x 48, $C_2^3:D_4$ x 48, $C_4^2:C_2^2$ x 48, $C_2^2:\OD_{16}$ x 40, $C_2^4\times C_4$ x 40, $C_2^2\times \OD_{16}$ x 37, $C_4^2:C_2^2$ x 36, $C_2^3:C_8$ x 24, $C_4^2:C_4$ x 16, $C_2^4:C_4$ x 10, $D_4\times C_2^3$ x 8, $C_2^3.D_4$ x 6, $C_2^2\times \SD_{16}$ x 4, $C_2^2\times C_4^2$ x 4, $C_2^3\times C_8$ x 3, $Q_8\times C_2^3$ x 2, $C_2^2\times Q_{16}$ x 2, $C_2^2\times D_8$ x 2, $C_2^6$

Order 32: $D_4:C_2^2$ x 368, $C_2^3\times C_4$ x 318, $C_4\times D_4$ x 256, $C_2^2\times D_4$ x 154, $Q_{16}:C_2$ x 128, $D_8:C_2$ x 128, $D_8:C_2$ x 128, $C_2^2.D_4$ x 128, $C_2^2:Q_8$ x 128, $C_4:D_4$ x 128, $C_2^2\wr C_2$ x 128, $C_2^3:C_4$ x 120, $C_2\times \OD_{16}$ x 110, $C_4^2:C_2$ x 80, $C_2^2.D_4$ x 72, $C_4\wr C_2$ x 64, $C_2\times \SD_{16}$ x 48, $C_2\times C_4^2$ x 36, $C_2^2\times Q_8$ x 34, $C_2^2:C_8$ x 32, $C_2^2\times C_8$ x 30, $C_2^5$ x 29, $C_2\times Q_{16}$ x 24, $C_2\times D_8$ x 24

Order 16: $C_2^2\times C_4$ x 672, $C_2\times D_4$ x 476, $D_4:C_2$ x 320, $C_2^4$ x 243, $C_2^2:C_4$ x 160, $C_4:C_4$ x 96, $C_2\times Q_8$ x 92, $\SD_{16}$ x 64, $\OD_{16}$ x 52, $C_2\times C_8$ x 48, $C_4^2$ x 40, $Q_{16}$ x 32, $D_8$ x 32

Order 8: $C_2^3$ x 535, $C_2\times C_4$ x 380, $D_4$ x 232, $Q_8$ x 40, $C_8$ x 12

Order 4: $C_2^2$ x 279, $C_4$ x 44

Order 2: $C_2$ x 35

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 512: $C_2^6.D_4$

Order 256: $D_8:C_2^4$, $C_4^2:C_2^4$, $C_4^2:C_2^4$, $C_2^4:\OD_{16}$, $C_2^5.D_4$, $C_2^5.D_4$

Order 128: $C_4^2:C_2^3$ x 4, $D_8:C_2^3$ x 3, $C_2^3\times \OD_{16}$ x 2, $C_4^2:C_2^3$ x 2, $C_2^3:\OD_{16}$ x 2, $C_4^2:D_4$, $C_2^4.D_4$, $D_4:C_2^4$, $C_2^5\times C_4$, $Q_{16}:C_2^3$, $C_8:C_2^4$, $D_8:C_2^3$, $C_2^4.D_4$, $C_2^4:Q_8$, $C_2^4:D_4$, $C_2^4:D_4$, $C_4^2:C_2^3$, $C_4^2:C_2^3$, $C_4^2:C_2^3$, $C_2^4:C_8$

Order 64: $C_2^4\times C_4$ x 13, $D_4:C_2^3$ x 12, $C_2^2\times \OD_{16}$ x 8, $C_4^2:C_2^2$ x 5, $D_8:C_2^2$ x 4, $C_4^2:C_2^2$ x 4, $D_4\times C_2^3$ x 3, $C_2^4:C_4$ x 3, $C_2^2:\OD_{16}$ x 2, $Q_{16}:C_2^2$ x 2, $C_8:C_2^3$ x 2, $D_8:C_2^2$ x 2, $C_2^3\times C_8$ x 2, $C_2^3.D_4$ x 2, $C_2^3:Q_8$ x 2, $C_2^3:D_4$ x 2, $C_2^3:D_4$ x 2, $C_4^2:C_2^2$ x 2, $C_2^3.D_4$ x 2, $C_4^2:C_2^2$ x 2, $C_2^3:C_8$, $C_2^6$, $Q_8\times C_2^3$, $C_2^2\times Q_{16}$, $C_2^2\times \SD_{16}$, $C_2^2\times D_8$, $C_2^2\times C_4^2$, $C_4^2:C_4$, $C_4^2:C_2^2$

Order 32: $C_2^3\times C_4$ x 52, $D_4:C_2^2$ x 31, $C_2^2\times D_4$ x 14, $C_2\times \OD_{16}$ x 12, $C_2^5$ x 8, $C_2^2\times Q_8$ x 6, $C_2^2\times C_8$ x 6, $C_4^2:C_2$ x 5, $C_2^3:C_4$ x 4, $C_2^2.D_4$ x 3, $Q_{16}:C_2$ x 2, $D_8:C_2$ x 2, $D_8:C_2$ x 2, $C_2^2.D_4$ x 2, $C_2^2:Q_8$ x 2, $C_4:D_4$ x 2, $C_2^2\wr C_2$ x 2, $C_4\times D_4$ x 2, $C_2\times C_4^2$ x 2, $C_2^2:C_8$, $C_2\times Q_{16}$, $C_2\times \SD_{16}$, $C_2\times D_8$, $C_4\wr C_2$

Order 16: $C_2^2\times C_4$ x 77, $C_2^4$ x 26, $D_4:C_2$ x 20, $C_2\times D_4$ x 20, $C_2\times Q_8$ x 9, $C_2\times C_8$ x 6, $\OD_{16}$ x 4, $C_2^2:C_4$ x 4, $C_4:C_4$ x 3, $C_4^2$ x 2, $Q_{16}$, $\SD_{16}$, $D_8$

Order 8: $C_2\times C_4$ x 45, $C_2^3$ x 36, $D_4$ x 9, $Q_8$ x 4, $C_8$ x 2

Order 4: $C_2^2$ x 26, $C_4$ x 9

Order 2: $C_2$ x 8

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

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

Order 256: $C_2^5.D_4$ ($C_2$) x 24, $C_4^2:C_2^4$ ($C_2$) x 2, $C_4^2:C_2^4$ ($C_2$) x 2, $D_8:C_2^4$ ($C_2$), $C_2^4:\OD_{16}$ ($C_2$), $C_2^5.D_4$ ($C_2$)

Order 128: $C_4^2:D_4$ ($C_2^2$) x 64, $C_4^2:C_2^3$ ($C_2^2$) x 24, $C_4^2:C_2^3$ ($C_2^2$) x 24, $C_2^4.D_4$ ($C_2^2$) x 12, $D_8:C_2^3$ ($C_2^2$) x 12, $C_2^3:\OD_{16}$ ($C_2^2$) x 12, $D_4:C_2^4$ ($C_2^2$) x 2, $C_2^3\times \OD_{16}$ ($C_2^2$) x 2, $C_4^2:C_2^3$ ($C_2^2$) x 2, $C_2^5\times C_4$ ($C_2^2$)

Order 64: $C_4^2:C_2^2$ ($C_2^3$) x 32, $C_4^2:C_2^2$ ($C_2^3$) x 32, $C_2^2:\OD_{16}$ ($C_2^3$) x 16, $D_8:C_2^2$ ($C_2^3$) x 16, $C_4^2:C_4$ ($C_2^3$) x 16, $D_4:C_2^3$ ($C_2^3$) x 12, $D_4:C_2^3$ ($D_4$) x 12, $C_2^2\times \OD_{16}$ ($C_2^3$) x 12, $C_4^2:C_2^2$ ($C_2^3$) x 12, $C_4^2:C_2^2$ ($D_4$) x 12, $C_2^4\times C_4$ ($C_2^3$) x 7, $C_2^4\times C_4$ ($D_4$) x 7, $C_2^2\times \OD_{16}$ ($D_4$) x 6, $C_2^2\times C_4^2$ ($D_4$) x 4, $Q_8\times C_2^3$ ($D_4$) x 2, $D_4\times C_2^3$ ($D_4$) x 2, $C_2^3\times C_8$ ($D_4$) x 2, $C_2^6$ ($D_4$)

Order 32: $C_2^3\times C_4$ ($C_2\times D_4$) x 41, $D_4:C_2^2$ ($C_2\times D_4$) x 24, $C_4^2:C_2$ ($C_2\times D_4$) x 24, $C_2\times C_4^2$ ($C_2\times D_4$) x 24, $C_2^2\times Q_8$ ($C_2\times D_4$) x 12, $C_2^2\times D_4$ ($C_2\times D_4$) x 12, $C_2\times \OD_{16}$ ($C_2\times D_4$) x 12, $C_2^2\times C_8$ ($C_2\times D_4$) x 12, $D_4:C_2^2$ ($C_2^4$) x 8, $C_2\times \OD_{16}$ ($C_2^4$) x 8, $C_4^2:C_2$ ($C_2^4$) x 8, $C_2^5$ ($C_2\times D_4$) x 7, $C_2^3\times C_4$ ($C_2^4$) x 7, $C_2^3\times C_4$ ($D_4:C_2$) x 4

Order 16: $C_2^2\times C_4$ ($C_4:D_4$) x 44, $C_2^2\times C_4$ ($C_2^2\wr C_2$) x 40, $C_2^2\times C_4$ ($C_2^2\times D_4$) x 37, $C_4^2$ ($C_2^2\times D_4$) x 16, $C_2^2\times C_4$ ($C_4:D_4$) x 16, $C_2\times C_8$ ($C_2^2\times D_4$) x 8, $C_2^4$ ($C_2^2\wr C_2$) x 8, $C_2\times Q_8$ ($C_2^2\times D_4$) x 8, $C_2\times D_4$ ($C_2^2\times D_4$) x 8, $C_2^4$ ($C_2^2\times D_4$) x 7, $C_2^2\times C_4$ ($D_4:C_2^2$) x 6, $C_2^4$ ($C_4:D_4$) x 4, $C_2^2\times C_4$ ($C_2^5$)

Order 8: $C_2\times C_4$ ($C_2^3:D_4$) x 48, $C_2\times C_4$ ($C_2^3:D_4$) x 30, $C_2\times C_4$ ($C_2^3:D_4$) x 24, $C_2^3$ ($C_2^3:D_4$) x 16, $C_2^3$ ($C_2^3:D_4$) x 12, $C_2\times C_4$ ($C_4^2:C_2^2$) x 12, $C_2^3$ ($C_2^3:D_4$) x 6, $C_2\times C_4$ ($D_4\times C_2^3$) x 5, $C_2^3$ ($D_4\times C_2^3$), $C_2\times C_4$ ($D_4:C_2^3$)

Order 4: $C_2^2$ ($C_2^4:D_4$) x 12, $C_4$ ($C_2^4:D_4$) x 12, $C_2^2$ ($C_4^2:D_4$) x 4, $C_2^2$ ($C_2^4:D_4$) x 2, $C_4$ ($C_2^4:D_4$) x 2, $C_2^2$ ($C_2^4:D_4$), $C_4$ ($C_4^2:C_2^3$), $C_4$ ($C_2^4:D_4$)

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

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: $D_8:C_2^4$ ($C_2$), $C_4^2:C_2^4$ ($C_2$), $C_4^2:C_2^4$ ($C_2$), $C_2^4:\OD_{16}$ ($C_2$), $C_2^5.D_4$ ($C_2$), $C_2^5.D_4$ ($C_2$)

Order 128: $C_2^3\times \OD_{16}$ ($C_2^2$) x 2, $C_4^2:D_4$ ($C_2^2$), $C_2^4.D_4$ ($C_2^2$), $D_4:C_2^4$ ($C_2^2$), $C_2^5\times C_4$ ($C_2^2$), $D_8:C_2^3$ ($C_2^2$), $C_4^2:C_2^3$ ($C_2^2$), $C_4^2:C_2^3$ ($C_2^2$), $C_4^2:C_2^3$ ($C_2^2$), $C_2^3:\OD_{16}$ ($C_2^2$)

Order 64: $C_2^4\times C_4$ ($C_2^3$) x 2, $C_2^4\times C_4$ ($D_4$) x 2, $C_2^2\times \OD_{16}$ ($C_2^3$) x 2, $C_2^2:\OD_{16}$ ($C_2^3$), $C_2^6$ ($D_4$), $D_4:C_2^3$ ($C_2^3$), $D_4:C_2^3$ ($D_4$), $Q_8\times C_2^3$ ($D_4$), $D_4\times C_2^3$ ($D_4$), $D_8:C_2^2$ ($C_2^3$), $C_2^2\times \OD_{16}$ ($D_4$), $C_2^3\times C_8$ ($D_4$), $C_4^2:C_2^2$ ($C_2^3$), $C_4^2:C_2^2$ ($C_2^3$), $C_4^2:C_2^2$ ($D_4$), $C_2^2\times C_4^2$ ($D_4$), $C_4^2:C_4$ ($C_2^3$), $C_4^2:C_2^2$ ($C_2^3$)

Order 32: $C_2^3\times C_4$ ($C_2\times D_4$) x 9, $C_2^5$ ($C_2\times D_4$) x 2, $C_2^3\times C_4$ ($C_2^4$) x 2, $C_2^3\times C_4$ ($D_4:C_2$) x 2, $C_2\times \OD_{16}$ ($C_2^4$) x 2, $D_4:C_2^2$ ($C_2^4$), $D_4:C_2^2$ ($C_2\times D_4$), $C_2^2\times Q_8$ ($C_2\times D_4$), $C_2^2\times D_4$ ($C_2\times D_4$), $C_2\times \OD_{16}$ ($C_2\times D_4$), $C_2^2\times C_8$ ($C_2\times D_4$), $C_4^2:C_2$ ($C_2^4$), $C_4^2:C_2$ ($C_2\times D_4$), $C_2\times C_4^2$ ($C_2\times D_4$)

Order 16: $C_2^2\times C_4$ ($C_2^2\times D_4$) x 8, $C_2^2\times C_4$ ($C_4:D_4$) x 8, $C_2^2\times C_4$ ($C_2^2\wr C_2$) x 7, $C_2^2\times C_4$ ($C_4:D_4$) x 4, $C_2^4$ ($C_2^2\times D_4$) x 2, $C_2^4$ ($C_4:D_4$) x 2, $C_2^4$ ($C_2^2\wr C_2$) x 2, $C_2^2\times C_4$ ($D_4:C_2^2$) x 2, $C_2\times C_8$ ($C_2^2\times D_4$), $C_4^2$ ($C_2^2\times D_4$), $C_2\times Q_8$ ($C_2^2\times D_4$), $C_2\times D_4$ ($C_2^2\times D_4$), $C_2^2\times C_4$ ($C_2^5$)

Order 8: $C_2\times C_4$ ($C_2^3:D_4$) x 7, $C_2\times C_4$ ($C_2^3:D_4$) x 6, $C_2\times C_4$ ($C_2^3:D_4$) x 5, $C_2^3$ ($C_2^3:D_4$) x 4, $C_2\times C_4$ ($D_4\times C_2^3$) x 3, $C_2\times C_4$ ($C_4^2:C_2^2$) x 3, $C_2^3$ ($C_2^3:D_4$) x 2, $C_2^3$ ($C_2^3:D_4$) x 2, $C_2^3$ ($D_4\times C_2^3$), $C_2\times C_4$ ($D_4:C_2^3$)

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

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

$C_2^6.D_4$

$\rhd$

$C_2^2\times C_4$

$\rhd$

$C_1$

Chief series

$C_2^6.D_4$

$\rhd$

$C_2^5.D_4$

$\rhd$

$C_2^3\times \OD_{16}$

$\rhd$

$C_2^4\times C_4$

$\rhd$

$C_2^3\times C_4$

$\rhd$

$C_2^2\times C_4$

$\rhd$

$C_2\times C_4$

$\rhd$

$C_2^2$

$\rhd$

$C_2$

$\rhd$

$C_1$

Lower central series

$C_2^6.D_4$

$\rhd$

$C_2^2\times C_4$

$\rhd$

$C_2$

$\rhd$

$C_1$

Upper central series

$C_1$

$\lhd$

$C_2^2\times C_4$

$\lhd$

$C_2^4\times C_4$

$\lhd$

$C_2^6.D_4$

Supergroups

This group is a maximal subgroup of 4 larger groups in the database.

This group is a maximal quotient of 0 larger groups in the database.

Character theory

Complex character table

See the $104 \times 104$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $92 \times 92$ rational character table.