#7624101

order	512
nilpotency class	4
isoclinism	isoclinic class of #7623613
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

1290 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72322566
maximal normal subgroup of	#72322566

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7531297);