#6573750

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

4529 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)(21,28)(22,37)(23,36)(24,38);(2,4)(9,10,13,15)(11,17,20,30)(12,23,25,36)(14,16,27,19)(18,29,31,26)(21,35,28,32)(22,34,37,40)(24,33,38,39);(6,8)(10,16)(11,21)(14,27)(15,19)(17,30)(18,23)(20,28)(22,38)(24,37)(26,29)(31,36)(33,39);(9,11,13,20)(10,17,15,30)(12,24,25,38)(14,28,27,21)(16,32,19,35)(18,34,31,40)(22,29,37,26)(23,33,36,39);(9,12,13,25)(10,18,15,31)(11,22,20,37)(14,29,27,26)(16,23,19,36)(17,33,30,39)(21,38,28,24)(32,40,35,34)

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,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, 6480946);