#6542702

order	512
nilpotency class	3
isoclinism	isoclinic class of #6533075
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	#62954	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄×D₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2948 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:

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

Polycyclic presentation:

2^9|a^2=fg,b^2=i,c^2=i,e^2=i,f^2=i,b^a=bf,c^a=cg,d^a=di,f^a=fi,g^a=gi,c^b=ch,f^b=fi,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, 6449898);