Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #2228038 |
| commutator, Frattini, cosocle | C25 | C25 |
| prime radical | #6435073 | C2 |
| LCS | G ▷ C25 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C25 ◁ G | |
| LECS | G ▷ C25 ▷ C2 ▷ C1 | |
4107 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 5 generators on 32 points:
(8,21)(12,26)(13,28)(20,24)(23,30)(31,32);(1,2)(3,11,14,7)(4,8)(5,9)(6,10)(12,19,26,22)(13,20,28,24)(15,23)(16,21)(17,30)(18,25)(27,32,29,31);(1,3)(2,7)(4,12)(5,13)(6,14)(8,22)(9,24)(10,11)(15,29)(16,26)(17,27)(18,28)(19,21)(20,25)(23,32)(30,31);(1,4)(2,8,10,21)(3,12)(5,17)(6,16)(7,19,11,22)(9,23,25,30)(13,27)(14,26)(15,18)(20,32,24,31)(28,29);(1,5)(2,9)(3,13,14,28)(4,15)(6,18)(7,20,11,24)(8,23)(10,25)(12,27,26,29)(16,17)(19,31,22,32)(21,30)Polycyclic presentation:
2^10|b^2=h,d^2=hi,e^2=h,b^a=bf,c^a=cg,c^b=chi,d^a=di,d^b=dghj,d^c=dhi,e^a=eh,e^b=eh,e^c=eghij,e^d=eij,f^d=fj,g^d=gj,g^e=gj,h^c=hj,i^b=ij,i^c=ij82 conjugacy classes — view the full character table.
| maximal subgroup of | #72237863, #72237925, #72251069, … |
| maximal normal subgroup of | #72237863, #72237925, #72251069, … |
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, 2, F.8);;
gap> SetPower(coll, 4, F.8*F.9);;
gap> SetPower(coll, 5, F.8);;
gap> SetConjugate(coll, 2, 1, F.2*F.6);;
gap> SetConjugate(coll, 3, 1, F.3*F.7);;
gap> SetConjugate(coll, 3, 2, F.3*F.8*F.9);;
gap> SetConjugate(coll, 4, 1, F.4*F.9);;
gap> SetConjugate(coll, 4, 2, F.4*F.7*F.8*F.10);;
gap> SetConjugate(coll, 4, 3, F.4*F.8*F.9);;
gap> SetConjugate(coll, 5, 1, F.5*F.8);;
gap> SetConjugate(coll, 5, 2, F.5*F.8);;
gap> SetConjugate(coll, 5, 3, F.5*F.7*F.8*F.9*F.10);;
gap> SetConjugate(coll, 5, 4, F.5*F.9*F.10);;
gap> SetConjugate(coll, 6, 4, F.6*F.10);;
gap> SetConjugate(coll, 7, 4, F.7*F.10);;
gap> SetConjugate(coll, 7, 5, F.7*F.10);;
gap> SetConjugate(coll, 8, 3, F.8*F.10);;
gap> SetConjugate(coll, 9, 2, F.9*F.10);;
gap> SetConjugate(coll, 9, 3, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);