Finite Group Databasebeta

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#72334261

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

Aliases

No aliases for this group.

Subgroups

At least 8668 conjugacy classes of subgroups; only the 2973 normal subgroups are shown.

Full subgroup data →

Presentation

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

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

Polycyclic presentation:

2^10|a^2=i,e^2=hj,b^a=bg,c^a=ci,c^b=cgh,d^a=dgh,e^a=ehij,e^b=egj,e^c=ei,e^d=egh,f^b=fghj,f^c=fi,f^d=fh,f^e=fgj

Conjugacy classes & character table

106 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.9);;
gap> SetPower(coll, 5, F.8*F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.7);;
gap> SetConjugate(coll, 3, 1, F.3*F.9);;
gap> SetConjugate(coll, 3, 2, F.3*F.7*F.8);;
gap> SetConjugate(coll, 4, 1, F.4*F.7*F.8);;
gap> SetConjugate(coll, 5, 1, F.5*F.8*F.9*F.10);;
gap> SetConjugate(coll, 5, 2, F.5*F.7*F.10);;
gap> SetConjugate(coll, 5, 3, F.5*F.9);;
gap> SetConjugate(coll, 5, 4, F.5*F.7*F.8);;
gap> SetConjugate(coll, 6, 2, F.6*F.7*F.8*F.10);;
gap> SetConjugate(coll, 6, 3, F.6*F.9);;
gap> SetConjugate(coll, 6, 4, F.6*F.8);;
gap> SetConjugate(coll, 6, 5, F.6*F.7*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);