#6586415

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

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=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, 6493611);