#6573512

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

4619 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

2^9|b^2=i,c^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, 6480708);