#7624883

order	512
nilpotency class	4
isoclinism	isoclinic class of #7624876
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂³×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁵
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄×D₄ ◁ C₂×C₈×D₄ ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1345 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)(3,7)(4,6)(5,8)(9,10)(11,16)(12,22)(13,26)(14,29)(15,19)(17,24)(18,27)(20,35)(21,34)(23,37)(25,38)(28,31)(30,36)(32,39)(33,40);(1,3)(2,5)(4,8)(6,7)(9,11,15,14)(10,16,19,29)(12,20,25,23)(13,21,28,27)(17,30,33,32)(18,26,34,31)(22,35,38,37)(24,36,40,39);(9,12,21,36,14,23,13,24,15,25,27,39,11,20,28,40)(10,17,31,37,16,30,18,22,19,33,26,35,29,32,34,38);(3,8)(5,7);(2,6)(5,7)(9,11,15,14)(10,16,19,29)(12,23,25,20)(13,21,28,27)(17,32,33,30)(18,26,34,31)(22,37,38,35)(24,39,40,36)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72314396, #72315829, #72315871
maximal normal subgroup of	#72314396, #72315829, #72315871

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7532079);