#6550585

order	512
nilpotency class	3
isoclinism	isoclinic class of #6533105
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×#121	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×C₄⋊C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1433 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)(3,7)(4,8)(5,6)(9,10)(11,15)(12,16)(13,14)(17,20)(18,22)(19,21)(23,24)(25,26,28,33,29,32,36,31)(27,30,34,39,35,38,40,37);(2,6)(7,8)(10,14)(11,17)(16,21)(18,23)(25,27,29,35)(26,30,32,38)(28,34,36,40)(31,37,33,39);(1,3,5,4)(2,7,6,8)(9,11,13,17)(10,15,14,20)(12,18,19,23)(16,22,21,24)(26,31)(28,36)(30,37)(32,33)(34,40)(38,39);(1,4,5,3)(2,8,6,7)(12,19)(16,21)(18,23)(22,24);(9,12,13,19)(10,16,14,21)(11,18,17,23)(15,22,20,24)(26,32)(30,38)(31,33)(37,39)

Polycyclic presentation:

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

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, 6457781);