#6827187

order	512
nilpotency class	3
isoclinism	isoclinic class of #6806839
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	#30722	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1446 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)(11,14)(12,16)(13,15)(17,18)(19,27)(20,31)(21,23)(22,26)(24,30)(25,33)(28,37)(29,38)(32,40)(34,36)(35,45)(39,43)(41,44)(42,48)(46,47);(9,11,14,10)(12,16,13,15)(18,23)(19,28)(20,32)(22,34)(24,37)(25,40)(27,39)(30,43)(31,45)(33,35)(38,41)(42,46);(2,5)(6,8)(9,12,14,13)(10,15,11,16)(17,19,22,28,21,30,34,43)(18,24,26,39,23,27,36,37)(20,29,35,42,33,44,32,46)(25,38,40,48,31,41,45,47);(1,3,4,7)(2,6,5,8)(9,12,14,13)(10,15,11,16)(18,23)(20,33)(24,27)(26,36)(29,44)(32,35)(37,39)(42,46);(9,13,14,12)(10,15,11,16)(17,20,34,32,21,33,22,35)(18,25,36,45,23,31,26,40)(19,29,43,46,30,44,28,42)(24,38,37,47,27,41,39,48)

Polycyclic presentation:

2^9|b^2=fi,c^2=fhi,d^2=gi,e^2=fh,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,c^b=ch,e^b=efh,h^b=hi,e^d=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, 6734383);