#6595869

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573671
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 ◁ C₂×C₄×D₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3122 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:

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

Polycyclic presentation:

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

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, 6503065);