#7625082

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

1168 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)(17,18)(19,25)(20,34)(21,27)(22,41)(23,43)(24,29)(26,38)(28,35)(30,40)(31,44)(32,39)(33,48)(36,47)(37,45)(42,46);(1,3)(2,6)(4,11)(5,10)(7,14)(8,9)(12,15)(13,16)(17,19)(18,25)(20,30)(21,26)(22,32)(23,33)(24,31)(27,38)(28,37)(29,44)(34,40)(35,45)(36,46)(39,41)(42,47)(43,48);(17,20,35,44,23,36,42,48,24,37,40,25,21,39,22,38)(18,26,41,32,27,19,30,45,29,33,46,47,43,31,28,34);(3,10)(6,9)(11,16)(14,15)(17,21,24,23)(18,27,29,43)(19,26,31,33)(20,36,37,39)(22,40,42,35)(25,38,44,48)(28,41,30,46)(32,34,47,45);(1,4,5,13)(2,7,8,12)(3,11,10,16)(6,14,9,15)(19,31)(20,37)(25,44)(26,33)(32,47)(34,45)(36,39)(38,48)

Polycyclic presentation:

2^9|a^2=f,c^2=gh,d^2=i,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, 7532278);