#6573522

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573511
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂¹⁰

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 ◁ #2388 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

4564 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)(3,4)(5,6)(7,8)(9,10,14,15)(11,16,22,19)(12,17,26,24)(13,18,29,30)(20,31,23,32)(21,27,34,35)(25,33,39,28)(36,37,38,40);(2,4)(9,11,14,22)(10,16,15,19)(12,23,26,20)(13,27,29,35)(17,32,24,31)(18,34,30,21)(25,37,39,40)(28,36,33,38);(6,8)(10,15)(11,19)(12,24)(16,22)(17,26)(18,30)(21,35)(25,28)(27,34)(31,32)(33,39)(37,40);(9,12,14,26)(10,17,15,24)(11,20,22,23)(13,28,29,33)(16,31,19,32)(18,25,30,39)(21,37,34,40)(27,38,35,36);(9,13,14,29)(10,18,15,30)(11,21,22,34)(12,25,26,39)(16,27,19,35)(17,33,24,28)(20,36,23,38)(31,37,32,40)

Polycyclic presentation:

2^9|a^2=i,b^2=i,d^2=i,e^2=i,h^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=di,e^b=ehi,h^b=hi,d^c=dhi,h^c=hi,e^d=eh,h^d=hi,h^e=hi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6480718);