#6624567

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573615
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2285 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1508 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:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=fg,d^2=fghi,e^2=hi,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,d^b=di,e^b=eh,f^b=fi,g^b=gi,d^c=dh,f^c=fi,g^c=gi,e^d=eh

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6531763);