#6573415

order	512
nilpotency class	3
isoclinism	isoclinic class of #6533141
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#120
commutator, Frattini, cosocle	C₂⁴	C₂⁵
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄⋊C₄ ◁ G
LECS	G ▷ C₂⁴ ▷ C₂ ▷ C₁

3248 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=h,c^2=h,d^2=h,e^2=hi,b^a=bf,c^a=cg,d^a=di,g^a=gi,c^b=ch,g^b=gi,f^c=fi,e^d=eh

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6480611);