#6760970

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667980
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×D₄	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1277 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)(17,18,21,27,22,26,23,28)(19,29,36,39,31,24,32,41)(20,33,37,25,35,42,34,40)(30,44,46,43,45,48,47,38);(1,2)(3,6)(4,8)(5,7)(9,10)(11,15)(12,14)(13,16)(17,19)(18,24)(20,30)(21,36)(22,31)(23,32)(25,38)(26,29)(27,41)(28,39)(33,48)(34,47)(35,45)(37,46)(40,43)(42,44);(9,11,12,16)(10,13,14,15)(17,20,22,35)(18,25,26,40)(19,30,31,45)(21,34,23,37)(24,38,29,43)(27,33,28,42)(32,46,36,47)(39,44,41,48);(1,3,5,4)(2,6,7,8)(18,26)(19,31)(25,40)(27,28)(30,45)(32,36)(33,42)(46,47);(1,4,5,3)(2,6,7,8)(10,14)(13,15)(17,21,22,23)(18,27,26,28)(19,32,31,36)(20,34,35,37)(24,39,29,41)(25,33,40,42)(30,46,45,47)(38,44,43,48)

Polycyclic presentation:

2^9|a^2=fg,c^2=hi,d^2=fi,e^2=f,g^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,d^b=di,e^b=efh,g^c=gi

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, 6668166);