#6596196

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573715
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	C₂⁴	C₂⁵
prime radical	#62991	C₂
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄⋊C₄ ◁ G
LECS	G ▷ C₂⁴ ▷ C₂ ▷ C₁

2994 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)(11,16)(12,24,27,17)(13,28)(14,18)(15,19)(20,29,35,37)(21,36)(22,30)(23,31)(25,32,26,38)(33,40,34,39);(1,2,4,5)(3,6,7,8)(9,11)(10,16)(12,20)(13,21)(14,22)(15,23)(17,29)(18,30)(19,31)(24,37)(25,33)(26,34)(27,35)(28,36)(32,39)(38,40);(2,5)(3,7)(9,12)(10,17)(11,20)(13,25)(14,26)(15,27)(16,29)(18,32)(19,24)(21,33)(22,34)(23,35)(28,38)(30,39)(31,37)(36,40);(1,3)(2,6)(4,7)(5,8)(9,13,15,14)(10,18,19,28)(11,21,23,22)(12,25,27,26)(16,30,31,36)(17,32,24,38)(20,33,35,34)(29,39,37,40);(2,5)(3,7)(9,14,15,13)(10,18,19,28)(11,22,23,21)(12,26,27,25)(16,30,31,36)(17,32,24,38)(20,34,35,33)(29,39,37,40)

Polycyclic presentation:

2^9|a^2=f,b^2=h,d^2=i,e^2=i,b^a=bf,c^a=cg,d^a=di,g^a=gi,c^b=ch,e^b=eh,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, 6503392);