#6558892

order	512
nilpotency class	3
isoclinism	isoclinic class of #6535664
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	C₂²×D₈	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2295 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1342 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)(4,6)(5,8)(17,18)(19,28)(20,32)(21,26)(22,38)(23,27)(24,31)(25,35)(29,42)(30,41)(33,39)(34,43)(36,47)(37,40)(44,45)(46,48);(1,3,5,4)(2,6,8,7)(9,10)(11,14)(12,13)(15,16)(17,19,22,31,23,30,21,37)(18,24,26,28,27,40,38,41)(20,33,35,48,36,29,34,45)(25,42,32,46,43,39,47,44);(10,13)(14,16)(17,20,21,34,23,36,22,35)(18,25,38,47,27,43,26,32)(19,29,37,48,30,33,31,45)(24,39,41,46,40,42,28,44);(1,3,5,4)(2,7,8,6)(9,11)(10,14)(12,15)(13,16)(19,30)(24,40)(28,41)(29,33)(31,37)(39,42)(44,46)(45,48);(1,4,5,3)(2,6,8,7)(11,15)(14,16)(17,21,23,22)(18,26,27,38)(19,31,30,37)(20,34,36,35)(24,41,40,28)(25,32,43,47)(29,45,33,48)(39,46,42,44)

Polycyclic presentation:

2^9|b^2=f,c^2=gi,d^2=fgi,e^2=fg,f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,c^b=ch,d^b=di,e^b=ei,e^d=ehi

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, 6466088);