#8689875

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

4610 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:

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

Polycyclic presentation:

2^9|b^2=gh,d^2=h,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,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, 8597071);