#6573457

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573451
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

4510 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)(11,16,22,31)(12,23,26,36)(13,15,27,18)(17,29,34,25)(20,35,28,30)(21,33,38,39)(24,32,37,40);(6,8)(10,15)(11,20)(13,27)(16,31)(17,23)(18,19)(21,37)(22,28)(24,38)(25,29)(32,40)(34,36);(9,11,14,22)(10,16,19,31)(12,24,26,37)(13,28,27,20)(15,30,18,35)(17,33,34,39)(21,29,38,25)(23,32,36,40);(9,12,14,26)(10,17,19,34)(11,21,22,38)(13,29,27,25)(15,23,18,36)(16,32,31,40)(20,37,28,24)(30,39,35,33)

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,e^b=eh,h^b=hi,d^c=dh,h^c=hi,e^d=eh,h^d=hi,h^e=hi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6480653);