#6596165

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573671
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	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄×D₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2933 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:

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

Polycyclic presentation:

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

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, 6503361);