#6696853

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667772
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	#23871	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2398 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1606 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)(10,14)(11,12)(17,18,22,26)(19,23,30,39)(20,24,32,41)(21,25,34,43)(27,36,29,38)(28,37,45,47)(31,40,46,48)(33,42,35,44);(9,10)(11,15)(12,16)(13,14)(17,19)(18,23)(20,31)(21,27)(22,30)(24,40)(25,36)(26,39)(28,33)(29,34)(32,46)(35,45)(37,42)(38,43)(41,48)(44,47);(2,5)(6,8)(9,11,13,12)(10,15,14,16)(19,27)(21,34)(23,36)(25,43)(28,31)(29,30)(33,35)(37,40)(38,39)(42,44)(45,46)(47,48);(1,3,4,7)(2,6,5,8)(9,12,13,11)(10,16,14,15)(20,32)(24,41)(28,45)(31,46)(33,35)(37,47)(40,48)(42,44);(10,14)(15,16)(17,20,22,32)(18,24,26,41)(19,28,30,45)(21,35,34,33)(23,37,39,47)(25,44,43,42)(27,31,29,46)(36,40,38,48)

Polycyclic presentation:

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

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, 6604049);