#6937311

order	512
nilpotency class	3
isoclinism	isoclinic class of #6884809
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
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄²×D₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1466 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)(25,26,30,31,29,33,27,34)(28,36,35,40,38,39,37,32);(1,2)(3,6)(4,5)(7,8)(10,14)(12,19)(15,20)(18,23)(25,27,29,30)(26,31,33,34)(28,37,38,35)(32,36,40,39);(2,5)(3,7)(9,11)(10,15)(12,18)(13,17)(14,20)(16,22)(19,23)(21,24)(25,28,29,38)(26,32,33,40)(27,35,30,37)(31,39,34,36);(1,3,4,7)(2,6,5,8)(10,14)(11,17)(16,21)(18,23)(26,33)(31,34)(32,40)(36,39);(2,5)(6,8)(9,12,13,19)(10,16,14,21)(11,18,17,23)(15,22,20,24)(25,27,29,30)(26,34,33,31)(28,35,38,37)(32,36,40,39)

Polycyclic presentation:

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

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6844507);