Abstract group 512.6276981: $C_2^5.C_4^2$

• Label: 512.6276981
• Order: \( 2^{9} \)
• Exponent: \( 2^{3} \)
• Nilpotent: yes
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{8} \)
• $\card{Z(G)}$: \( 2^{7} \)
• $\card{\Aut(G)}$: \( 2^{26} \cdot 3 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{24} \cdot 3 \)
• Perm deg.: $20$
• Trans deg.: $256$
• Rank: $5$

Group information

Description:	$C_2^5.C_4^2$
Order:	\(512\)\(\medspace = 2^{9} \)
Exponent:	\(8\)\(\medspace = 2^{3} \)
Automorphism group:	$C_2^{15}.C_2^6.D_6.C_2^3$, of order \(201326592\)\(\medspace = 2^{26} \cdot 3 \)
Composition factors:	$C_2$ x 9
Nilpotency class:	$2$
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	63	192	256	512
Conjugacy classes	1	47	144	128	320
Divisions	1	47	72	32	152
Autjugacy classes	1	7	5	1	14

Dimension	1	2	4
Irr. complex chars.	256	64	0	320
Irr. rational chars.	32	64	56	152

Minimal presentations

Permutation degree:	$20$
Transitive degree:	$256$
Rank:	$5$
Inequivalent generating 5-tuples:	$52080$

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:	${\langle a, b, c, d, e, f \mid a^{8}=b^{4}=c^{2}=d^{2}=e^{2}=f^{2}=[a,c]= \!\cdots\! \rangle}$
Permutation group:	Degree $20$ $\langle(1,2)(3,4)(5,6,9,8)(7,10,12,11)(13,14)(15,16)(17,18,19,20), (1,3,4,2)(5,7,8,11,9,12,6,10) \!\cdots\! \rangle$
Direct product:	$C_2$ ${}^2$ $\, \times\, $ $C_4$ $\, \times\, $ $(C_2^2:C_8)$
Semidirect product:	$(C_2^4\times C_4)$ $\,\rtimes\,$ $C_8$	$C_2^4$ $\,\rtimes\,$ $(C_4\times C_8)$	$(C_2^4.C_4^2)$ $\,\rtimes\,$ $C_2$	$(C_2^3\times C_4\times C_8)$ $\,\rtimes\,$ $C_2$	all 15
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$C_2^5$ . $C_4^2$	$(C_2^5:C_8)$ . $C_2$	$(C_2^5\times C_4)$ . $C_4$	$C_2^6$ . $(C_2\times C_4)$	all 88

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

Homology

Abelianization:	$C_{2}^{3} \times C_{4} \times C_{8} $
Schur multiplier:	$C_{2}^{10} \times C_{4}$
Commutator length:	$1$

Subgroups

There are 14210 subgroups in 8890 conjugacy classes, 3570 normal (18 characteristic).

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

Special subgroups

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

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

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: $C_2^5.C_4^2$

Order 256: $C_2^4.C_4^2$ x 24, $C_2^5:C_8$ x 2, $C_2^3\times C_4\times C_8$ x 2, $C_2^4.C_4^2$ x 2, $C_2^4\times C_4^2$

Order 128: $C_4^2.D_4$ x 64, $C_2^4:C_8$ x 56, $C_2^3\times C_4^2$ x 44, $C_2^2\times C_4\times C_8$ x 40, $C_2^3.C_4^2$ x 24, $C_2^4\times C_8$ x 4, $C_2^5\times C_4$ x 3

Order 64: $C_2^2\times C_4^2$ x 344, $C_2^3:C_8$ x 224, $C_2^4\times C_4$ x 138, $C_2\times C_4\times C_8$ x 128, $C_2^3\times C_8$ x 88, $C_2^2.C_4^2$ x 32, $C_2^6$

Order 32: $C_2^3\times C_4$ x 1164, $C_2\times C_4^2$ x 560, $C_2^2\times C_8$ x 336, $C_2^2:C_8$ x 128, $C_4\times C_8$ x 64, $C_2^5$ x 47

Order 16: $C_2^2\times C_4$ x 2208, $C_2^4$ x 411, $C_2\times C_8$ x 256, $C_4^2$ x 160

Order 8: $C_2\times C_4$ x 936, $C_2^3$ x 835, $C_8$ x 32

Order 4: $C_2^2$ x 411, $C_4$ x 72

Order 2: $C_2$ x 47

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 512: $C_2^5.C_4^2$

Order 256: $C_2^4\times C_4^2$, $C_2^5:C_8$, $C_2^3\times C_4\times C_8$, $C_2^4.C_4^2$, $C_2^4.C_4^2$

Order 128: $C_2^3\times C_4^2$ x 6, $C_2^5\times C_4$ x 2, $C_2^4:C_8$ x 2, $C_2^2\times C_4\times C_8$ x 2, $C_4^2.D_4$, $C_2^3.C_4^2$, $C_2^4\times C_8$

Order 64: $C_2^4\times C_4$ x 13, $C_2^2\times C_4^2$ x 12, $C_2^3\times C_8$ x 4, $C_2^3:C_8$ x 2, $C_2\times C_4\times C_8$ x 2, $C_2^6$, $C_2^2.C_4^2$

Order 32: $C_2^3\times C_4$ x 31, $C_2\times C_4^2$ x 12, $C_2^5$ x 7, $C_2^2\times C_8$ x 5, $C_2^2:C_8$, $C_4\times C_8$

Order 16: $C_2^2\times C_4$ x 36, $C_2^4$ x 17, $C_4^2$ x 5, $C_2\times C_8$ x 4

Order 8: $C_2\times C_4$ x 23, $C_2^3$ x 22, $C_8$

Order 4: $C_2^2$ x 17, $C_4$ x 5

Order 2: $C_2$ x 7

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 512: $C_2^5.C_4^2$ ($C_1$)

Order 256: $C_2^4.C_4^2$ ($C_2$) x 24, $C_2^5:C_8$ ($C_2$) x 2, $C_2^3\times C_4\times C_8$ ($C_2$) x 2, $C_2^4.C_4^2$ ($C_2$) x 2, $C_2^4\times C_4^2$ ($C_2$)

Order 128: $C_4^2.D_4$ ($C_2^2$) x 64, $C_2^4:C_8$ ($C_4$) x 32, $C_2^3.C_4^2$ ($C_2^2$) x 24, $C_2^4:C_8$ ($C_2^2$) x 24, $C_2^2\times C_4\times C_8$ ($C_2^2$) x 24, $C_2^3\times C_4^2$ ($C_2^2$) x 14, $C_2^3\times C_4^2$ ($C_4$) x 14, $C_2^4\times C_8$ ($C_2^2$) x 4, $C_2^5\times C_4$ ($C_4$) x 2, $C_2^5\times C_4$ ($C_2^2$)

Order 64: $C_2^3:C_8$ ($C_2\times C_4$) x 192, $C_2^2\times C_4^2$ ($C_2\times C_4$) x 84, $C_2^4\times C_4$ ($C_2\times C_4$) x 51, $C_2^3:C_8$ ($C_2^3$) x 32, $C_2\times C_4\times C_8$ ($C_2^3$) x 32, $C_2^4\times C_4$ ($C_8$) x 32, $C_2^3\times C_8$ ($C_2\times C_4$) x 32, $C_2^2.C_4^2$ ($C_2^3$) x 32, $C_2^2\times C_4^2$ ($C_2^3$) x 28, $C_2^3\times C_8$ ($C_2^3$) x 24, $C_2^2\times C_4^2$ ($D_4$) x 16, $C_2^4\times C_4$ ($C_2^3$) x 7, $C_2^6$ ($C_2\times C_4$)

Order 32: $C_2^3\times C_4$ ($C_2\times C_8$) x 224, $C_2^3\times C_4$ ($C_2^2\times C_4$) x 133, $C_2^2:C_8$ ($C_2^2\times C_4$) x 128, $C_2^2\times C_8$ ($C_2^2\times C_4$) x 96, $C_2^3\times C_4$ ($C_4^2$) x 56, $C_2\times C_4^2$ ($C_2^2\times C_4$) x 56, $C_2\times C_4^2$ ($C_2^2:C_4$) x 48, $C_2\times C_4^2$ ($C_2\times D_4$) x 48, $C_2^5$ ($C_2\times C_8$) x 16, $C_2^3\times C_4$ ($\OD_{16}$) x 16, $C_2^3\times C_4$ ($C_2^2:C_4$) x 16, $C_2^2\times C_8$ ($C_2^4$) x 16, $C_2^5$ ($C_4^2$) x 8, $C_2^3\times C_4$ ($D_4:C_2$) x 8, $C_2^3\times C_4$ ($C_2\times D_4$) x 8, $C_2\times C_4^2$ ($C_2^4$) x 8, $C_2^5$ ($C_2^2\times C_4$) x 7, $C_2^3\times C_4$ ($C_2^4$) x 7

Order 16: $C_2^2\times C_4$ ($C_2^2\times C_8$) x 224, $C_2^2\times C_4$ ($C_2^2:C_8$) x 128, $C_2^2\times C_4$ ($C_2\times C_4^2$) x 84, $C_2^4$ ($C_4\times C_8$) x 64, $C_2^2\times C_4$ ($C_4\times D_4$) x 64, $C_2^2\times C_4$ ($C_2^3:C_4$) x 60, $C_2^4$ ($C_2^2\times C_8$) x 56, $C_2^2\times C_4$ ($C_2^3\times C_4$) x 51, $C_4^2$ ($C_2^3:C_4$) x 48, $C_2^2\times C_4$ ($C_2\times \OD_{16}$) x 48, $C_2\times C_8$ ($C_2^3\times C_4$) x 32, $C_2^4$ ($C_2\times C_4^2$) x 28, $C_4^2$ ($C_2^2\times D_4$) x 16, $C_2^2\times C_4$ ($D_4:C_2^2$) x 12, $C_2^2\times C_4$ ($C_2^2\times D_4$) x 12, $C_2^2\times C_4$ ($C_4^2:C_2$) x 12, $C_2^4$ ($C_2\times \OD_{16}$) x 8, $C_2^4$ ($C_2^3\times C_4$) x 7, $C_2^4$ ($C_4^2:C_2$) x 4, $C_2^4$ ($C_2^3:C_4$) x 4, $C_2^2\times C_4$ ($C_2^5$)

Order 8: $C_2\times C_4$ ($C_2^3:C_8$) x 192, $C_2^3$ ($C_2\times C_4\times C_8$) x 112, $C_2\times C_4$ ($C_2^2:C_4^2$) x 48, $C_2\times C_4$ ($C_4^2:C_2^2$) x 48, $C_2^3$ ($C_2^3:C_8$) x 32, $C_2^3$ ($C_4^2.C_4$) x 32, $C_2\times C_4$ ($C_2^3\times C_8$) x 32, $C_2^3$ ($C_2^3\times C_8$) x 28, $C_2\times C_4$ ($C_2^4:C_4$) x 22, $C_2^3$ ($C_4\times \OD_{16}$) x 16, $C_2^3$ ($C_2^2:C_4^2$) x 16, $C_2\times C_4$ ($C_2^2\times \OD_{16}$) x 16, $C_2^3$ ($C_2^2\times C_4^2$) x 14, $C_2\times C_4$ ($C_2^2\times C_4^2$) x 14, $C_2^3$ ($C_2^2\times \OD_{16}$) x 12, $C_2^3$ ($C_4^2:C_2^2$) x 6, $C_2^3$ ($C_2^4:C_4$) x 6, $C_2\times C_4$ ($C_4^2:C_2^2$) x 6, $C_2\times C_4$ ($D_4:C_2^3$) x 2, $C_2\times C_4$ ($D_4\times C_2^3$) x 2, $C_2\times C_4$ ($C_2^4\times C_4$) x 2, $C_2^3$ ($C_2^4\times C_4$)

Order 4: $C_2^2$ ($C_4^2.D_4$) x 64, $C_4$ ($C_2^4:C_8$) x 32, $C_2^2$ ($C_2^2\times C_4\times C_8$) x 28, $C_2^2$ ($C_4^2.C_2^3$) x 24, $C_2^2$ ($C_2^4:C_8$) x 24, $C_2^2$ ($C_2^3.C_4^2$) x 12, $C_2^2$ ($C_2^3:C_4^2$) x 12, $C_4$ ($C_2^3:C_4^2$) x 12, $C_4$ ($C_4^2:C_2^3$) x 4, $C_2^2$ ($C_2^3\times \OD_{16}$) x 2, $C_2^2$ ($C_2^4\times C_8$) x 2, $C_2^2$ ($C_4^2:C_2^3$), $C_2^2$ ($C_2^5:C_4$), $C_2^2$ ($C_2^3\times C_4^2$)

Order 2: $C_2$ ($C_2^4.C_4^2$) x 24, $C_2$ ($C_4^2.C_2^4$) x 2, $C_2$ ($C_2^5:C_8$) x 2, $C_2$ ($C_4^2.C_2^4$), $C_2$ ($C_2^3\times C_4\times C_8$), $C_2$ ($C_2^4:C_4^2$)

Order 1: $C_1$ ($C_2^5.C_4^2$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 512: $C_2^5.C_4^2$ ($C_1$)

Order 256: $C_2^4\times C_4^2$ ($C_2$), $C_2^5:C_8$ ($C_2$), $C_2^3\times C_4\times C_8$ ($C_2$), $C_2^4.C_4^2$ ($C_2$), $C_2^4.C_4^2$ ($C_2$)

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

Order 64: $C_2^4\times C_4$ ($C_2\times C_4$) x 6, $C_2^2\times C_4^2$ ($C_2\times C_4$) x 4, $C_2^2\times C_4^2$ ($C_2^3$) x 3, $C_2^4\times C_4$ ($C_2^3$) x 2, $C_2^3:C_8$ ($C_2^3$), $C_2^3:C_8$ ($C_2\times C_4$), $C_2\times C_4\times C_8$ ($C_2^3$), $C_2^6$ ($C_2\times C_4$), $C_2^4\times C_4$ ($C_8$), $C_2^3\times C_8$ ($C_2^3$), $C_2^3\times C_8$ ($C_2\times C_4$), $C_2^2\times C_4^2$ ($D_4$), $C_2^2.C_4^2$ ($C_2^3$)

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

Order 16: $C_2^2\times C_4$ ($C_2^3\times C_4$) x 6, $C_2^2\times C_4$ ($C_2^2\times C_8$) x 3, $C_2^2\times C_4$ ($C_2\times C_4^2$) x 3, $C_2^4$ ($C_2^3\times C_4$) x 2, $C_2^4$ ($C_2^2\times C_8$) x 2, $C_2^4$ ($C_2\times C_4^2$) x 2, $C_2^2\times C_4$ ($C_2^3:C_4$) x 2, $C_2\times C_8$ ($C_2^3\times C_4$), $C_4^2$ ($C_2^2\times D_4$), $C_4^2$ ($C_2^3:C_4$), $C_2^4$ ($C_2\times \OD_{16}$), $C_2^4$ ($C_4\times C_8$), $C_2^4$ ($C_4^2:C_2$), $C_2^4$ ($C_2^3:C_4$), $C_2^2\times C_4$ ($C_2^5$), $C_2^2\times C_4$ ($C_2^2:C_8$), $C_2^2\times C_4$ ($D_4:C_2^2$), $C_2^2\times C_4$ ($C_2^2\times D_4$), $C_2^2\times C_4$ ($C_2\times \OD_{16}$), $C_2^2\times C_4$ ($C_4\times D_4$), $C_2^2\times C_4$ ($C_4^2:C_2$)

Order 8: $C_2^3$ ($C_2\times C_4\times C_8$) x 2, $C_2^3$ ($C_2^3\times C_8$) x 2, $C_2^3$ ($C_2^2\times C_4^2$) x 2, $C_2\times C_4$ ($C_2^3\times C_8$) x 2, $C_2\times C_4$ ($C_2^4:C_4$) x 2, $C_2\times C_4$ ($C_2^2\times C_4^2$) x 2, $C_2^3$ ($C_2^3:C_8$), $C_2^3$ ($C_4\times \OD_{16}$), $C_2^3$ ($C_2^2:C_4^2$), $C_2^3$ ($C_2^4\times C_4$), $C_2^3$ ($C_2^2\times \OD_{16}$), $C_2^3$ ($C_4^2:C_2^2$), $C_2^3$ ($C_2^4:C_4$), $C_2^3$ ($C_4^2.C_4$), $C_2\times C_4$ ($C_2^3:C_8$), $C_2\times C_4$ ($C_2^2:C_4^2$), $C_2\times C_4$ ($D_4:C_2^3$), $C_2\times C_4$ ($D_4\times C_2^3$), $C_2\times C_4$ ($C_2^4\times C_4$), $C_2\times C_4$ ($C_2^2\times \OD_{16}$), $C_2\times C_4$ ($C_4^2:C_2^2$), $C_2\times C_4$ ($C_4^2:C_2^2$)

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

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

Order 1: $C_1$ ($C_2^5.C_4^2$)

Series

Derived series

$C_2^5.C_4^2$

$\rhd$

$C_2$

$\rhd$

$C_1$

Chief series

$C_2^5.C_4^2$

$\rhd$

$C_2^4\times C_4^2$

$\rhd$

$C_2^5\times C_4$

$\rhd$

$C_2^4\times C_4$

$\rhd$

$C_2^3\times C_4$

$\rhd$

$C_2^2\times C_4$

$\rhd$

$C_2^3$

$\rhd$

$C_2^2$

$\rhd$

$C_2$

$\rhd$

$C_1$

Lower central series

$C_2^5.C_4^2$

$\rhd$

$C_2$

$\rhd$

$C_1$

Upper central series

$C_1$

$\lhd$

$C_2^3\times C_4^2$

$\lhd$

$C_2^5.C_4^2$

Character theory

Complex character table

See the $320 \times 320$ character table (warning: may be slow to load). Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $152 \times 152$ rational character table (warning: may be slow to load).