#6500231

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

3141 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:

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

Polycyclic presentation:

2^9|a^2=f,c^2=i,d^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, 6407427);