#7624026

order	512
nilpotency class	4
isoclinism	isoclinic class of #7623604
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 ◁ #529 ◁ #2966 ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1319 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72260392
maximal normal subgroup of	#72260392

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7531222);