#6828635

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
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1483 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)(3,6)(4,5)(7,8)(9,10)(11,15)(12,14)(13,16)(17,18,21,23)(19,24,29,34)(20,25,32,35)(22,26,28,36)(27,39,41,37)(30,38,43,46)(31,44,33,40)(42,48,45,47);(9,11)(10,13)(12,16)(14,15)(18,23)(19,27)(20,31)(22,28)(24,37)(25,40)(29,41)(32,33)(34,39)(35,44)(38,46)(42,45);(2,5)(6,8)(10,14)(11,16)(17,19,28,41,21,29,22,27)(18,24,36,37,23,34,26,39)(20,30,31,45,32,43,33,42)(25,38,44,47,35,46,40,48);(1,3,4,7)(2,6,5,8)(10,14)(13,15)(18,23)(19,29)(25,35)(26,36)(27,41)(30,43)(40,44)(42,45);(17,20,28,31,21,32,22,33)(18,25,36,44,23,35,26,40)(19,30,41,45,29,43,27,42)(24,38,37,47,34,46,39,48)

Polycyclic presentation:

2^9|a^2=i,c^2=fh,d^2=fgi,e^2=fh,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, 6735831);