#6832948

order	512
nilpotency class	3
isoclinism	isoclinic class of #6806647
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 ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1294 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=fi,c^2=fhi,d^2=i,e^2=ghi,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,d^c=di,e^c=ei

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6740144);