#7625081

order	512
nilpotency class	4
isoclinism	isoclinic class of #7624888
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

1186 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 48 points:

(1,2,5,8)(3,9,10,6)(4,12,13,7)(11,14,16,15)(18,24)(19,28)(20,23)(22,29)(25,38)(27,43)(30,44)(31,46)(33,45)(34,35)(36,41)(37,39)(40,48)(42,47);(1,3)(2,6)(4,11)(5,10)(7,14)(8,9)(12,15)(13,16)(17,18,21,24)(19,25,30,36)(20,32,23,26)(22,27,35,37)(28,41,44,38)(29,39,34,43)(31,42,33,40)(45,47,46,48);(17,19,29,45,23,31,35,44,21,30,34,46,20,33,22,28)(18,25,39,47,26,42,37,38,24,36,43,48,32,40,27,41);(3,10)(6,9)(11,16)(14,15)(18,24)(19,30)(26,32)(27,37)(28,44)(31,33)(39,43)(45,46);(1,4,5,13)(2,7,8,12)(3,11,10,16)(6,14,9,15)(17,20,21,23)(18,26,24,32)(19,31,30,33)(22,34,35,29)(25,40,36,42)(27,39,37,43)(28,45,44,46)(38,47,41,48)

Polycyclic presentation:

2^9|a^2=fi,b^2=i,c^2=gh,e^2=f,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,d^c=di,e^c=ei

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7532277);