#6586341

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573511
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2402 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1277 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6493537);