#6572931

order	512
nilpotency class	3
isoclinism	isoclinic class of #6533123
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 ◁ #2283 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1316 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:

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

Polycyclic presentation:

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

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, 6480127);