#6586421

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573451
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

1272 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)(17,18,22,27)(19,29,31,24)(20,25,37,45)(21,26,35,43)(23,28,36,44)(30,47,48,39)(32,42,34,40)(33,46,38,41);(2,6)(7,8)(17,19,22,31)(18,24,27,29)(20,34,37,32)(21,30,35,48)(23,38,36,33)(25,42,45,40)(26,39,43,47)(28,46,44,41);(1,3,5,4)(2,7,6,8)(9,11,12,16)(10,13,14,15)(17,20)(18,25)(19,30)(21,36)(22,37)(23,35)(24,39)(26,44)(27,45)(28,43)(29,47)(31,48)(32,33)(34,38)(40,41)(42,46);(1,4,5,3)(2,8,6,7)(19,31)(20,35)(21,37)(23,36)(24,29)(25,43)(26,45)(28,44)(30,34)(32,48)(39,42)(40,47);(17,21,22,35)(18,26,27,43)(19,32,31,34)(20,36,37,23)(24,40,29,42)(25,44,45,28)(30,33,48,38)(39,41,47,46)

Polycyclic presentation:

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

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, 6493617);