#6597315

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	C₂×#548	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2293 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1687 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=fhi,c^2=i,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, 6504511);