Finite Group Databasebeta

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#72304080

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

Aliases

No aliases for this group.

Subgroups

At least 7596 conjugacy classes of subgroups; only the 3077 normal subgroups are shown.

Full subgroup data →

Presentation

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

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

Polycyclic presentation:

2^10|a^2=h,b^2=j,e^2=i,d^a=di,d^b=dg,d^c=dgh,e^a=ehi,e^b=ej,e^c=egi,e^d=ei,f^a=fi,f^d=fgh,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, 5, F.9);;
gap> SetConjugate(coll, 4, 1, F.4*F.9);;
gap> SetConjugate(coll, 4, 2, F.4*F.7);;
gap> SetConjugate(coll, 4, 3, F.4*F.7*F.8);;
gap> SetConjugate(coll, 5, 1, F.5*F.8*F.9);;
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.9);;
gap> SetConjugate(coll, 6, 1, F.6*F.9);;
gap> SetConjugate(coll, 6, 4, F.6*F.7*F.8);;
gap> SetConjugate(coll, 6, 5, F.6*F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);