#6500229

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	#62874	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×#41 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3192 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^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, 6407425);