#6573530

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573523
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

4524 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)(10,15)(11,20)(16,19)(18,31)(22,24)(23,37)(25,33)(26,39);(2,4)(9,10,13,15)(11,21,20,17)(12,25,27,33)(14,16,29,19)(18,28,31,30)(22,36,24,32)(23,35,37,40)(26,34,39,38);(6,8)(10,16)(11,22)(14,29)(15,19)(18,33)(20,24)(23,26)(25,31)(28,30)(32,36)(35,40)(37,39);(9,11,13,20)(10,17,15,21)(12,26,27,39)(14,22,29,24)(16,32,19,36)(18,35,31,40)(23,30,37,28)(25,38,33,34);(9,12,13,27)(10,18,15,31)(11,23,20,37)(14,30,29,28)(16,33,19,25)(17,34,21,38)(22,26,24,39)(32,40,36,35)

Polycyclic presentation:

2^9|b^2=i,d^2=i,e^2=i,h^2=i,b^a=bf,c^a=cg,d^a=di,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

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6480726);