#6596197

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
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄⋊C₄ ◁ G
LECS	G ▷ C₂⁴ ▷ C₂ ▷ C₁

3044 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)(11,20)(12,25,28,17)(13,29)(14,18)(15,19)(16,24)(21,33,37,30)(22,31)(23,34)(26,32,27,38)(35,39,36,40);(1,2,4,5)(3,6,7,8)(9,11)(10,16)(12,21)(13,22)(14,23)(15,24)(17,30)(18,31)(19,20)(25,33)(26,35)(27,36)(28,37)(29,34)(32,39)(38,40);(2,5)(3,7)(9,12)(10,17)(11,21)(13,26)(14,27)(15,28)(16,30)(18,32)(19,25)(20,33)(22,35)(23,36)(24,37)(29,38)(31,39)(34,40);(1,3)(2,6)(4,7)(5,8)(9,13,15,14)(10,18,19,29)(11,22,24,23)(12,26,28,27)(16,31,20,34)(17,32,25,38)(21,35,37,36)(30,39,33,40);(2,5)(3,7)(9,14,15,13)(10,18,19,29)(11,23,24,22)(12,27,28,26)(16,31,20,34)(17,32,25,38)(21,36,37,35)(30,39,33,40)

Polycyclic presentation:

2^9|a^2=fi,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, 6503393);