#6581249

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	C₂×#121	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2401 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1363 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)(17,18,22,26)(19,23,29,39)(20,30,34,24)(21,25,35,43)(27,37,36,44)(28,41,32,38)(31,42,33,40)(45,48,46,47);(2,6)(7,8)(9,11,12,15)(10,13,14,16)(19,27)(20,31)(21,35)(23,37)(24,40)(25,43)(29,36)(30,42)(33,34)(39,44)(45,46)(47,48);(1,3,5,4)(2,7,6,8)(10,14)(13,16)(17,19)(18,23)(20,32)(21,36)(22,29)(24,41)(25,44)(26,39)(27,35)(28,34)(30,38)(31,46)(33,45)(37,43)(40,48)(42,47);(1,4,5,3)(2,8,6,7)(19,27)(20,33)(21,35)(23,37)(24,42)(25,43)(28,32)(29,36)(30,40)(31,34)(38,41)(39,44);(17,20,22,34)(18,24,26,30)(19,28,29,32)(21,33,35,31)(23,38,39,41)(25,42,43,40)(27,45,36,46)(37,47,44,48)

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=f,d^2=f,e^2=i,h^2=i,b^a=bf,c^a=cg,e^a=ei,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, 6488445);