#6502334

order	512
nilpotency class	3
isoclinism	isoclinic class of #6499804
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	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2296 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1610 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)(3,8)(4,6)(5,7)(17,18,23,27)(19,24,30,40)(20,31,34,42)(21,26,22,36)(25,32,41,33)(28,43,45,47)(29,38,35,39)(37,46,48,44);(1,3,5,4)(2,6,7,8)(9,10)(11,15)(12,14)(13,16)(17,19)(18,24)(20,28)(21,35)(22,29)(23,30)(25,37)(26,39)(27,40)(31,43)(32,46)(33,44)(34,45)(36,38)(41,48)(42,47);(9,11)(10,13)(12,16)(14,15)(17,20,23,34)(18,25,27,41)(19,28,30,45)(21,33,22,32)(24,37,40,48)(26,42,36,31)(29,46,35,44)(38,43,39,47);(11,16)(13,15)(17,21,23,22)(18,26,27,36)(19,29,30,35)(20,32,34,33)(24,38,40,39)(25,31,41,42)(28,44,45,46)(37,47,48,43);(1,4,5,3)(2,6,7,8)(10,14)(13,15)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6409530);