#6832935

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

Polycyclic presentation:

2^9|a^2=gi,b^2=f,c^2=fh,d^2=i,e^2=gh,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, 6740131);