#6596342

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
prime radical	#62993	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄×D₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2909 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=h,c^2=i,d^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=ei,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, 6503538);