#6500233

order	512
nilpotency class	3
isoclinism	isoclinic class of #6500226
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
prime radical	#62875	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×#41 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3117 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:

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

Polycyclic presentation:

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

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, 6407429);