#6623426

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573591
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	C₂×#121	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#533 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1477 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)(17,18,21,26,23,28,22,27)(19,29,36,24,31,41,32,39)(20,33,37,25,35,42,34,40)(30,38,45,43,44,47,46,48);(1,3,5,4)(2,6,7,8)(10,14)(12,15)(17,19,23,31)(18,24,28,39)(20,30,35,44)(21,32,22,36)(25,38,40,47)(26,29,27,41)(33,43,42,48)(34,45,37,46);(1,4,5,3)(2,6,7,8)(9,10)(11,12)(13,14)(15,16)(17,20)(18,25)(19,30)(21,34)(22,37)(23,35)(24,38)(26,33)(27,42)(28,40)(29,43)(31,44)(32,45)(36,46)(39,47)(41,48);(9,11,13,16)(10,15,14,12)(19,31)(24,39)(29,41)(30,44)(32,36)(38,47)(43,48)(45,46);(9,12,13,15)(10,11,14,16)(17,21,23,22)(18,26,28,27)(19,32,31,36)(20,34,35,37)(24,29,39,41)(25,33,40,42)(30,45,44,46)(38,43,47,48)

Polycyclic presentation:

2^9|a^2=fi,b^2=fg,c^2=fgi,d^2=h,e^2=hi,f^2=i,g^2=i,b^a=bf,c^a=cg,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

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6530622);