Finite Group Databasebeta

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#72318298

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

Aliases

No aliases for this group.

Subgroups

3886 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

Polycyclic presentation:

2^10|a^2=g,b^2=i,d^2=i,b^a=bh,c^b=cf,d^a=dfgi,d^b=df,e^a=eghj,e^b=egi,e^c=ehj,f^e=fj,g^d=gj,h^b=hj,i^a=ij

Conjugacy classes & character table

76 conjugacy classes — view the full character table.

Relationships

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