#6912200

order	512
nilpotency class	3
isoclinism	isoclinic class of #6907909
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×#537	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2297 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1473 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=gi,d^2=i,e^2=fi,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,e^b=efhi,d^c=dfh

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, 6819396);