#6820310

order	512
nilpotency class	3
isoclinism	isoclinic class of #6806623
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×D₈	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2399 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1327 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6727506);