Finite Group Databasebeta

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#72328933

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

Aliases

No aliases for this group.

Subgroups

At least 7606 conjugacy classes of subgroups; only the 1043 normal subgroups are shown.

Full subgroup data →

Presentation

A minimal faithful permutation representation — 5 generators on 32 points:

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

Polycyclic presentation:

2^10|a^2=j,b^2=i,d^2=hij,e^2=gj,b^a=bg,c^a=cj,c^b=cf,d^a=dh,d^b=dhj,d^c=dh,e^a=efh,e^b=egij,e^c=efhj,e^d=efghij

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