Finite Group Databasebeta

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#72304091

• p-group
• nilpotent
• supersolvable
• metabelian
• solvable
• monomial
• stem

Aliases

No aliases for this group.

Subgroups

At least 7355 conjugacy classes of subgroups; only the 3089 normal subgroups are shown.

Full subgroup data →

Presentation

A minimal faithful permutation representation — 6 generators on 32 points:

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

Polycyclic presentation:

2^10|a^2=h,b^2=j,d^2=hi,d^a=dhi,d^b=dg,d^c=dghi,e^a=eh,e^b=ej,e^c=egi,e^d=ehi,f^a=fi,f^d=fghi,f^e=fij

Conjugacy classes & character table

142 conjugacy classes — view the full character table.

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from its polycyclic presentation:

gap> F := FreeGroup(IsSyllableWordsFamily, "a", "b", "c", "d", "e", "f", "g", "h", "i", "j");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 1, F.8);;
gap> SetPower(coll, 2, F.10);;
gap> SetPower(coll, 4, F.8*F.9);;
gap> SetConjugate(coll, 4, 1, F.4*F.8*F.9);;
gap> SetConjugate(coll, 4, 2, F.4*F.7);;
gap> SetConjugate(coll, 4, 3, F.4*F.7*F.8*F.9);;
gap> SetConjugate(coll, 5, 1, F.5*F.8);;
gap> SetConjugate(coll, 5, 2, F.5*F.10);;
gap> SetConjugate(coll, 5, 3, F.5*F.7*F.9);;
gap> SetConjugate(coll, 5, 4, F.5*F.8*F.9);;
gap> SetConjugate(coll, 6, 1, F.6*F.9);;
gap> SetConjugate(coll, 6, 4, F.6*F.7*F.8*F.9);;
gap> SetConjugate(coll, 6, 5, F.6*F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);