#7624030

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

1315 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)(9,10,12,17)(11,20,21,15)(13,16,24,31)(14,27,25,19)(18,33,32,26)(22,34,37,28)(23,30,38,36)(29,39,35,40);(1,2)(3,6)(4,5)(7,8)(9,11,12,21)(10,15,17,20)(13,22,24,37)(14,23,25,38)(16,28,31,34)(18,29,32,35)(19,30,27,36)(26,39,33,40);(10,16)(13,26)(14,25)(15,28)(17,31)(18,19)(20,34)(22,39)(23,38)(24,33)(27,32)(29,30)(35,36)(37,40);(2,5)(6,8)(11,21)(15,20)(22,37)(23,38)(28,34)(29,35)(30,36)(39,40);(1,3,4,7)(2,6,5,8)(11,21)(15,20)(22,37)(23,38)(28,34)(29,35)(30,36)(39,40)

Polycyclic presentation:

2^9|a^2=i,b^2=i,e^2=fi,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	#72286697, #72320126, #72320155, …
maximal normal subgroup of	#72286697, #72320126, #72320155, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7531226);