#9141663

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

4680 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=gh,c^2=gi,d^2=gh,f^2=gh,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, 9048859);