#6500228

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

3114 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:

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

Polycyclic presentation:

2^9|a^2=f,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, 6407424);