#6583366

order	512
nilpotency class	3
isoclinism	isoclinic class of #6574086
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×D₄²
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#135	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1636 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=f,c^2=g,g^2=i,h^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=di,e^b=ehi,h^b=hi,d^c=dhi,h^c=hi,e^d=eh,h^d=hi,h^e=hi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6490562);