#6509433

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	C₂×#548	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1486 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)(19,24)(20,32)(21,26)(22,36)(23,27)(25,35)(28,44)(29,39)(30,40)(31,46)(33,48)(34,43)(37,42)(38,45)(41,47);(2,5)(6,8)(9,11)(10,14)(12,15)(13,16)(17,19,23,29)(18,24,27,39)(20,33,34,28)(21,30,22,31)(25,41,42,38)(26,40,36,46)(32,48,43,44)(35,47,37,45);(10,12)(11,16)(17,20)(18,25)(19,28)(21,35)(22,37)(23,34)(24,38)(26,43)(27,42)(29,33)(30,45)(31,47)(32,36)(39,41)(40,48)(44,46);(1,3,4,7)(2,6,5,8)(9,12,13,10)(11,15,16,14)(19,29)(20,34)(24,39)(25,42)(30,31)(32,43)(35,37)(40,46);(11,16)(14,15)(17,21,23,22)(18,26,27,36)(19,30,29,31)(20,35,34,37)(24,40,39,46)(25,43,42,32)(28,45,33,47)(38,48,41,44)

Polycyclic presentation:

2^9|a^2=hi,b^2=i,d^2=f,e^2=i,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=di,e^b=ehi,d^c=dh,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, 6416629);