Finite Group Databasebeta

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#72280628

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

Aliases

No aliases for this group.

Subgroups

20833 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

Polycyclic presentation:

2^10|a^2=g,b^2=i,c^2=h,e^2=gi,f^2=g,c^a=ch,c^b=chi,d^a=di,d^b=di,d^c=dgh,e^a=egh,e^b=ej,e^c=eh,e^d=egh,f^b=fi,f^e=fg

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.7);;
gap> SetPower(coll, 2, F.9);;
gap> SetPower(coll, 3, F.8);;
gap> SetPower(coll, 5, F.7*F.9);;
gap> SetPower(coll, 6, F.7);;
gap> SetConjugate(coll, 3, 1, F.3*F.8);;
gap> SetConjugate(coll, 3, 2, F.3*F.8*F.9);;
gap> SetConjugate(coll, 4, 1, F.4*F.9);;
gap> SetConjugate(coll, 4, 2, F.4*F.9);;
gap> SetConjugate(coll, 4, 3, F.4*F.7*F.8);;
gap> SetConjugate(coll, 5, 1, F.5*F.7*F.8);;
gap> SetConjugate(coll, 5, 2, F.5*F.10);;
gap> SetConjugate(coll, 5, 3, F.5*F.8);;
gap> SetConjugate(coll, 5, 4, F.5*F.7*F.8);;
gap> SetConjugate(coll, 6, 2, F.6*F.9);;
gap> SetConjugate(coll, 6, 5, F.6*F.7);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);