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#72318843

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

Aliases

No aliases for this group.

Subgroups

2901 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

Polycyclic presentation:

2^10|a^2=h,b^2=i,d^2=i,b^a=bhi,c^b=cf,d^a=dfg,d^b=dfi,e^a=egj,e^b=eghij,e^c=eh,f^e=fj,g^b=gj,g^d=gj,h^b=hj,i^a=ij

Conjugacy classes & character table

70 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, 2, F.9);;
gap> SetPower(coll, 4, F.9);;
gap> SetConjugate(coll, 2, 1, F.2*F.8*F.9);;
gap> SetConjugate(coll, 3, 2, F.3*F.6);;
gap> SetConjugate(coll, 4, 1, F.4*F.6*F.7);;
gap> SetConjugate(coll, 4, 2, F.4*F.6*F.9);;
gap> SetConjugate(coll, 5, 1, F.5*F.7*F.10);;
gap> SetConjugate(coll, 5, 2, F.5*F.7*F.8*F.9*F.10);;
gap> SetConjugate(coll, 5, 3, F.5*F.8);;
gap> SetConjugate(coll, 6, 5, F.6*F.10);;
gap> SetConjugate(coll, 7, 2, F.7*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);