Finite Group Databasebeta

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#72328635

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

Aliases

No aliases for this group.

Subgroups

At least 9273 conjugacy classes of subgroups; only the 3045 normal subgroups are shown.

Full subgroup data →

Presentation

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

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

Polycyclic presentation:

2^10|a^2=h,e^2=ghi,f^2=h,b^a=bhj,c^b=cg,d^a=dhi,d^b=dhi,e^a=egi,e^b=eghi,e^d=ehi,f^d=fh,f^e=fij

Conjugacy classes & character table

118 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.8);;
gap> SetPower(coll, 5, F.7*F.8*F.9);;
gap> SetPower(coll, 6, F.8);;
gap> SetConjugate(coll, 2, 1, F.2*F.8*F.10);;
gap> SetConjugate(coll, 3, 2, F.3*F.7);;
gap> SetConjugate(coll, 4, 1, F.4*F.8*F.9);;
gap> SetConjugate(coll, 4, 2, F.4*F.8*F.9);;
gap> SetConjugate(coll, 5, 1, F.5*F.7*F.9);;
gap> SetConjugate(coll, 5, 2, F.5*F.7*F.8*F.9);;
gap> SetConjugate(coll, 5, 4, F.5*F.8*F.9);;
gap> SetConjugate(coll, 6, 4, F.6*F.8);;
gap> SetConjugate(coll, 6, 5, F.6*F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);