#6595871

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

3100 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=i,d^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, 6503067);