#6507940

order	512
nilpotency class	3
isoclinism	isoclinic class of #6500456
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	C₂⁵	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2280 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1548 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=gi,e^2=f,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,e^b=ehi,d^c=dhi,e^d=ei

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, 6415136);