#6509311

order	512
nilpotency class	3
isoclinism	isoclinic class of #6499792
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×#547	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1477 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)(9,10,13,12)(11,15,16,14)(17,18)(19,28)(20,32)(21,26)(22,27)(23,37)(24,30)(25,35)(29,45)(31,41)(33,48)(34,44)(36,40)(38,43)(39,46)(42,47);(2,5)(6,8)(9,11)(10,14)(12,15)(13,16)(17,19,22,30)(18,24,27,28)(20,33,34,29)(21,36,23,31)(25,42,43,39)(26,41,37,40)(32,45,44,48)(35,46,38,47);(10,12)(11,16)(17,20,22,34)(18,25,27,43)(19,29,30,33)(21,35,23,38)(24,39,28,42)(26,44,37,32)(31,46,36,47)(40,48,41,45);(1,3,4,7)(2,6,5,8)(9,12,13,10)(11,15,16,14)(19,30)(20,34)(24,28)(25,43)(31,36)(32,44)(35,38)(40,41);(11,16)(14,15)(17,21,22,23)(18,26,27,37)(19,31,30,36)(20,35,34,38)(24,40,28,41)(25,44,43,32)(29,46,33,47)(39,48,42,45)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6416507);