#7624040

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

1297 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)(9,10,13,18)(11,20,23,38)(12,17,15,25)(14,28,27,39)(16,32,21,24)(19,26,35,30)(22,37,31,36)(29,33,40,34);(1,2)(3,6)(4,5)(7,8)(9,10)(11,21)(12,25)(13,18)(14,19)(15,17)(16,23)(20,32)(22,33)(24,38)(26,39)(27,35)(28,30)(29,37)(31,34)(36,40);(9,11)(10,16)(12,22)(13,23)(14,29)(15,31)(17,33)(18,21)(19,36)(20,28)(24,30)(25,34)(26,32)(27,40)(35,37)(38,39);(2,5)(6,8)(9,12,13,15)(10,17,18,25)(11,22,23,31)(14,26,27,30)(16,33,21,34)(19,28,35,39)(20,37,38,36)(24,29,32,40);(1,3,4,7)(2,6,5,8)(9,12,13,15)(10,17,18,25)(11,22,23,31)(14,26,27,30)(16,33,21,34)(19,28,35,39)(20,37,38,36)(24,29,32,40)

Polycyclic presentation:

2^9|a^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,c^b=ci,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	#72322566
maximal normal subgroup of	#72322566

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7531236);