#8759156

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

4416 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8666352);