#7624730

order	512
nilpotency class	4
isoclinism	isoclinic class of #7624150
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

1212 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:

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

Polycyclic presentation:

2^9|a^2=f,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,g^c=gh,h^c=hi

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7531926);