#7625057

order	512
nilpotency class	4
isoclinism	isoclinic class of #7624842
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

1161 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)(27,42)(30,43)(31,45)(32,44)(34,35)(36,41)(37,39)(40,48)(46,47);(1,3)(2,6)(4,11)(5,10)(7,14)(8,9)(12,15)(13,16)(17,18)(19,25)(20,33)(21,24)(22,27)(23,26)(28,41)(29,39)(30,36)(31,46)(32,40)(34,42)(35,37)(38,43)(44,47)(45,48);(17,19,29,45,23,32,35,43,21,30,34,44,20,31,22,28)(18,25,39,48,26,40,37,38,24,36,42,47,33,46,27,41);(3,10)(6,9)(11,16)(14,15)(18,24)(19,30)(26,33)(27,37)(28,43)(31,32)(39,42)(44,45);(1,4,5,13)(2,7,8,12)(3,11,10,16)(6,14,9,15)(17,20,21,23)(18,26,24,33)(19,31,30,32)(22,34,35,29)(25,40,36,46)(27,39,37,42)(28,44,43,45)(38,47,41,48)

Polycyclic presentation:

2^9|a^2=fi,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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7532253);