#6597615

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573591
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	#38903	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#529 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1599 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)(17,18,23,27)(19,24,31,39)(20,32,34,42)(21,28,22,26)(25,36,40,33)(29,43,45,47)(30,41,35,38)(37,44,48,46);(2,5)(6,8)(9,10,12,13)(11,14,15,16)(17,19,23,31)(18,24,27,39)(20,29,34,45)(21,35,22,30)(25,37,40,48)(26,41,28,38)(32,43,42,47)(33,46,36,44);(1,3,4,7)(2,6,5,8)(10,13)(11,15)(17,20,23,34)(18,25,27,40)(19,29,31,45)(21,33,22,36)(24,37,39,48)(26,32,28,42)(30,44,35,46)(38,47,41,43);(9,11)(10,14)(12,15)(13,16)(17,21,23,22)(18,26,27,28)(19,30,31,35)(20,33,34,36)(24,38,39,41)(25,32,40,42)(29,44,45,46)(37,47,48,43);(10,13)(11,15)(18,27)(19,31)(25,40)(26,28)(29,45)(30,35)(32,42)(44,46)

Polycyclic presentation:

2^9|a^2=i,b^2=hi,c^2=fi,d^2=i,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=ei,g^a=gi,c^b=ch,d^b=di,e^b=ehi,d^c=dh,g^c=gi,e^d=eh

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6504811);