#7625130

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

1239 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

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

Polycyclic presentation:

2^9|a^2=i,c^2=fghi,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,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, 7532326);