#6573743

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

4579 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)(10,15)(12,24)(16,30)(18,19)(20,35)(21,38)(25,39)(26,27);(2,4)(9,10)(11,20)(12,17)(13,15)(14,18)(16,23)(19,28)(21,37)(22,35)(24,31)(25,40)(26,34)(27,36)(29,30)(32,39)(33,38);(6,8)(9,11,13,22)(10,16,15,30)(12,25,24,39)(14,29,28,23)(17,33,31,37)(18,20,19,35)(21,26,38,27)(32,36,40,34);(9,12,13,24)(10,17,15,31)(11,21,22,38)(14,26,28,27)(16,32,30,40)(18,34,19,36)(20,37,35,33)(23,39,29,25);(10,18)(12,26)(14,28)(15,19)(16,20)(21,39)(23,29)(24,27)(25,38)(30,35)(32,40)(34,36)

Polycyclic presentation:

2^9|c^2=i,d^2=i,h^2=i,b^a=bf,c^a=cg,d^a=di,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, 6480939);