#6573516

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

4519 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);(2,4)(9,10,14,19)(11,20,24,16)(12,25,28,32)(13,15,29,18)(17,27,35,30)(21,36,23,31)(22,34,38,40)(26,33,39,37);(6,8)(10,15)(11,21)(13,29)(17,32)(18,19)(22,26)(23,24)(25,35)(27,30)(31,36)(34,40)(38,39);(9,11,14,24)(10,16,19,20)(12,26,28,39)(13,21,29,23)(15,31,18,36)(17,34,35,40)(22,30,38,27)(25,37,32,33);(9,12,14,28)(10,17,19,35)(11,22,24,38)(13,30,29,27)(15,32,18,25)(16,33,20,37)(21,26,23,39)(31,40,36,34)

Polycyclic presentation:

2^9|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, 6480712);