#6595868

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

3131 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)(11,21,24,33)(12,25,27,34)(13,20,14,19)(16,28,31,23)(17,29,32,26)(22,30,38,40)(35,36,39,37);(1,2,4,5)(3,6,7,8)(9,11)(10,16)(12,22)(13,28)(14,23)(15,24)(17,30)(18,31)(19,33)(20,21)(25,39)(26,37)(27,38)(29,36)(32,40)(34,35);(2,5)(3,7)(9,12,15,27)(10,17,18,32)(11,22,24,38)(13,26,14,29)(16,30,31,40)(19,25,20,34)(21,35,33,39)(23,36,28,37);(1,3)(2,6)(4,7)(5,8)(10,18)(16,31)(17,32)(19,20)(21,33)(25,34)(30,40)(35,39);(2,5)(3,7)

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^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, 6503064);