#9125951

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

4389 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)(11,14)(12,17)(13,18)(15,16)(19,24)(20,22)(21,23)(25,26,31,35)(27,36,38,32)(28,33,37,30)(29,34,39,40);(1,2)(3,6)(4,7)(5,8)(9,11)(10,14)(12,20)(13,21)(15,19)(16,24)(17,22)(18,23)(25,27,31,38)(26,32,35,36)(28,29,37,39)(30,34,33,40);(1,3)(2,5)(4,8)(6,7)(10,16)(11,19)(12,21)(18,22)(25,28,31,37)(26,33,35,30)(27,29,38,39)(32,40,36,34);(3,8)(5,6)(9,12,15,21)(10,17,16,23)(11,20,19,13)(14,22,24,18)(25,28)(26,30)(27,29)(31,37)(32,34)(33,35)(36,40)(38,39);(1,3)(2,6)(4,8)(5,7)(9,13,15,20)(10,18,16,22)(11,12,19,21)(14,17,24,23)(25,29)(26,34)(27,37)(28,38)(30,36)(31,39)(32,33)(35,40);(1,2)(3,5)(4,7)(6,8)(9,14,15,24)(10,19,16,11)(12,22,21,18)(13,23,20,17)(25,30)(26,28)(27,34)(29,32)(31,33)(35,37)(36,39)(38,40)

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=g,d^2=h,e^2=h,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^d=egi,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, 9033147);