#6690587

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667604
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	#35737	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1772 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)(3,7)(4,6)(5,8)(9,10)(11,14)(12,16)(13,15)(17,18,22,26)(19,23,28,38)(20,31,33,24)(21,25,35,43)(27,37,30,40)(29,46,45,39)(32,48,47,41)(34,44,36,42);(1,3,4,8)(2,5,6,7)(9,11,13,12)(10,14,15,16)(17,19)(18,23)(20,32)(21,27)(22,28)(24,41)(25,37)(26,38)(29,34)(30,35)(31,48)(33,47)(36,45)(39,42)(40,43)(44,46);(9,12,13,11)(10,14,15,16)(19,27)(21,35)(23,37)(25,43)(28,30)(29,32)(34,36)(38,40)(39,41)(42,44)(45,47)(46,48);(2,6)(5,7)(19,28)(20,33)(23,38)(24,31)(27,30)(34,36)(37,40)(42,44);(3,8)(5,7)(17,20,22,33)(18,24,26,31)(19,29,28,45)(21,36,35,34)(23,39,38,46)(25,44,43,42)(27,32,30,47)(37,41,40,48)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6597783);