#7623733

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 ◁ #525 ◁ #2929 ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1532 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:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=i,e^2=i,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	#72322867
maximal normal subgroup of	#72322867

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7530929);