Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #650612 |
| commutator, Frattini, cosocle | C25 | C25 |
| LCS | G ▷ C25 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C25 ◁ G | |
| LECS | G ▷ C25 ▷ C2 ▷ C1 | |
5986 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,6,10)(3,11)(4,8,16,22)(5,9,18,24)(7,14)(12,19)(13,23)(15,21,29,32)(17,20)(25,27)(26,30)(28,31);(1,3)(2,7)(4,12)(5,17)(6,14)(8,19)(9,23)(10,11)(13,18)(15,26)(16,27)(20,24)(21,30)(22,25)(28,29)(31,32);(1,4)(2,8)(3,12)(5,15)(6,16)(7,19)(9,21)(10,22)(11,25)(13,28)(14,27)(17,26)(18,29)(20,31)(23,30)(24,32);(1,5,6,18)(2,9,10,24)(3,13)(4,15)(7,20)(8,21)(11,23)(12,26,27,28)(14,17)(16,29)(19,30,25,31)(22,32);(4,16)(8,22)(15,29)(21,32)Polycyclic presentation:
2^10|a^2=i,d^2=h,b^a=bf,c^a=cij,d^b=dg,d^c=dgh,e^b=ehi,e^c=ei,f^a=fj,f^c=fj,g^c=gj,g^d=gj,h^b=hj,h^c=hj,i^b=ij106 conjugacy classes — view the full character table.
| maximal subgroup of | #72237804, #72251138, #72251302, … |
| maximal normal subgroup of | #72237804, #72251138, #72251302, … |
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.9);;
gap> SetPower(coll, 4, F.8);;
gap> SetConjugate(coll, 2, 1, F.2*F.6);;
gap> SetConjugate(coll, 3, 1, F.3*F.9*F.10);;
gap> SetConjugate(coll, 4, 2, F.4*F.7);;
gap> SetConjugate(coll, 4, 3, F.4*F.7*F.8);;
gap> SetConjugate(coll, 5, 2, F.5*F.8*F.9);;
gap> SetConjugate(coll, 5, 3, F.5*F.9);;
gap> SetConjugate(coll, 6, 1, F.6*F.10);;
gap> SetConjugate(coll, 6, 3, F.6*F.10);;
gap> SetConjugate(coll, 7, 3, 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, 8, 3, F.8*F.10);;
gap> SetConjugate(coll, 9, 2, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);