#8739259

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4297 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=hi,c^2=g,d^2=gh,e^2=gh,f^2=gh,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8646455);