#6818102

order	512
nilpotency class	3
isoclinism	isoclinic class of #6806839
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	C₂⁴
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2402 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1196 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6725298);