#6761290

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667900
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×D₄	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2293 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1273 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)(3,6)(4,8)(5,7)(17,18,21,26,22,27,34,24)(19,23,30,35,28,37,43,38)(20,25,33,42,32,41,46,39)(29,36,44,47,31,40,45,48);(1,3,6,2)(4,8,5,7)(9,10)(11,15)(12,14)(13,16)(17,19)(18,23)(20,31)(21,30)(22,28)(24,38)(25,40)(26,35)(27,37)(29,32)(33,45)(34,43)(36,41)(39,47)(42,48)(44,46);(9,11,12,16)(10,13,14,15)(18,24)(19,28)(21,34)(23,35)(25,39)(26,27)(29,31)(33,46)(36,47)(37,38)(40,48)(41,42);(1,4,6,5)(2,7,3,8)(17,20,22,32)(18,25,27,41)(19,29,28,31)(21,33,34,46)(23,36,37,40)(24,39,26,42)(30,44,43,45)(35,47,38,48);(1,5,6,4)(2,7,3,8)(10,14)(13,15)(19,28)(20,32)(23,37)(25,41)(30,43)(33,46)(35,38)(39,42)

Polycyclic presentation:

2^9|a^2=g,b^2=f,c^2=hi,d^2=fi,e^2=f,g^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,d^b=di,e^b=efh,g^c=gi,e^d=ei

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6668486);