#6596168

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	#63006	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄×D₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2966 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=h,d^2=i,e^2=i,f^2=i,b^a=bf,c^a=cg,d^a=di,g^a=gi,c^b=ch,e^b=eh,f^b=fi,g^b=gi,d^c=dh,f^c=fi,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, 6503364);