#6836894

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	#42800	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1612 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)(11,16)(12,14)(13,15)(17,18)(19,29)(20,25)(21,27)(22,26)(23,28)(24,31)(30,44)(32,42)(33,47)(34,40)(35,39)(36,41)(37,43)(38,46)(45,48);(1,2,4,5)(3,6,7,8)(17,19,22,31)(18,24,26,29)(20,33,34,35)(21,37,32,23)(25,39,40,47)(27,28,42,43)(30,45,46,36)(38,48,44,41);(9,11,13,12)(10,14,15,16)(17,20,23,36,22,34,37,45)(18,25,28,41,26,40,43,48)(19,30,32,33,31,46,21,35)(24,38,27,39,29,44,42,47);(1,3,4,7)(2,6,5,8)(18,26)(20,34)(24,29)(27,42)(28,43)(30,46)(33,35)(36,45);(2,5)(6,8)(9,12,13,11)(10,14,15,16)(17,21,37,31,22,32,23,19)(18,27,43,24,26,42,28,29)(20,35,45,46,34,33,36,30)(25,39,48,38,40,47,41,44)

Polycyclic presentation:

2^9|b^2=f,c^2=gh,d^2=fi,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

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, 6744090);