#7625112

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

1232 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)(3,10)(4,8)(5,14)(6,9)(7,12)(11,16)(13,15)(17,18,23,27)(19,24,29,39)(20,32,25,33)(21,35,37,34)(22,38,36,26)(28,42,40,43)(30,47,45,44)(31,48,46,41);(1,3)(2,7)(4,11)(5,13)(6,12)(8,15)(9,10)(14,16)(17,19)(18,24)(20,28)(21,30)(22,31)(23,29)(25,40)(26,41)(27,39)(32,42)(33,43)(34,44)(35,47)(36,46)(37,45)(38,48);(1,4,6,5)(2,8,9,14)(3,11,12,13)(7,15,10,16)(17,20,33,18,22,34,21,26,23,25,32,27,36,35,37,38)(19,28,43,24,31,44,30,41,29,40,42,39,46,47,45,48);(3,12)(7,10)(11,13)(15,16);(1,5,6,4)(2,8,9,14)(3,13,12,11)(7,15,10,16)(19,29)(24,39)(28,40)(30,45)(31,46)(41,48)(42,43)(44,47)

Polycyclic presentation:

2^9|a^2=i,c^2=fghi,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

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, 7532308);