#7624998

order	512
nilpotency class	4
isoclinism	isoclinic class of #7624824
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

1425 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)(3,7)(9,10,13,18)(11,20,23,35)(12,17,15,26)(14,29,28,40)(16,30,32,24)(19,27,33,21)(22,34,25,39)(31,37,36,38);(1,2)(3,6)(4,5)(7,8)(9,10)(11,16)(12,26)(13,18)(14,19)(15,17)(20,24)(21,29)(22,36)(23,32)(25,31)(27,40)(28,33)(30,35)(34,37)(38,39);(9,11,21,37,15,25,28,30,13,23,27,38,12,22,14,24)(10,16,29,34,17,31,33,35,18,32,40,39,26,36,19,20);(2,5)(6,8)(9,12,13,15)(10,17,18,26)(11,22,23,25)(14,27,28,21)(16,31,32,36)(19,29,33,40)(20,34,35,39)(24,38,30,37);(1,3,4,7)(2,6,5,8)(10,18)(16,32)(17,26)(19,33)(20,35)(29,40)(31,36)(34,39)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7532194);