#6581111

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	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2401 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1350 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)(3,7)(4,8)(5,6)(9,10)(11,13)(12,14)(15,16)(17,18)(19,24)(20,25)(21,23)(22,26)(27,40)(28,41)(29,36)(30,44)(31,43)(32,42)(33,37)(34,38)(35,39)(45,48)(46,47);(2,6)(7,8)(9,11,12,15)(10,13,14,16)(18,23)(19,27)(20,30)(22,34)(24,39)(25,42)(29,35)(32,33)(36,40)(37,44)(41,43)(45,46);(1,3,5,4)(2,7,6,8)(10,14)(13,16)(17,19)(18,24)(20,31)(21,29)(22,35)(23,36)(25,43)(26,39)(27,34)(28,33)(30,46)(32,45)(37,41)(38,40)(42,48)(44,47);(1,4,5,3)(2,8,6,7)(19,27)(20,32)(22,34)(24,40)(25,42)(26,38)(28,31)(29,35)(30,33)(36,39)(37,44)(41,43);(17,20,21,33)(18,25,23,37)(19,28,29,31)(22,32,34,30)(24,41,36,43)(26,42,38,44)(27,45,35,46)(39,47,40,48)

Polycyclic presentation:

2^9|b^2=g,c^2=fi,d^2=fi,e^2=i,h^2=i,b^a=bf,c^a=cg,c^b=ch,e^b=eh,h^b=hi,d^c=dhi,e^c=ei,h^c=hi,e^d=ehi,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, 6488307);