#8659362

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

4571 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=gh,b^2=h,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,f^d=fgi,f^e=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8566558);