#7623257

order	512
nilpotency class	4
isoclinism	isoclinic class of #33605
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁵
socle	C₂²	C₂³×D₈
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1423 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 40 points:

(1,2)(3,6)(4,5)(7,8)(9,10)(11,16)(12,22)(13,27)(14,29)(15,19)(17,25)(18,23)(20,33)(21,32)(24,35)(26,39)(28,40)(30,34)(31,38)(36,37);(2,5)(6,8)(9,11,15,14)(10,16,19,29)(12,20,24,26)(13,21,28,23)(17,30,31,36)(18,27,32,40)(22,33,35,39)(25,34,38,37);(9,12,23,37,14,26,28,38,15,24,21,34,11,20,13,25)(10,17,27,33,16,30,32,35,19,31,40,39,29,36,18,22);(1,3,4,7)(2,6,5,8);(12,24)(17,31)(20,26)(22,35)(25,38)(30,36)(33,39)(34,37)

Polycyclic presentation:

2^9|b^2=i,c^2=gh,d^2=f,g^2=hi,h^2=i,b^a=bf,c^a=cg,g^a=gh,h^a=hi,e^c=ei

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72287840, #72287844, #72287855, …
maximal normal subgroup of	#72287840, #72287844, #72287855, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7530453);