Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #32898 |
| commutator, Frattini, cosocle | C2×#117 | C24 |
| LCS | G ▷ C2×#117 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×#535 ◁ G | |
| LECS | G ▷ C2×#117 ▷ C23 ▷ C2 ▷ C1 | |
4143 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 40 points:
(2,5)(3,8)(6,7)(9,10,13,19)(11,21,24,17)(12,26,27,31)(14,20,28,36)(15,18,29,16)(22,33,23,35)(25,37,39,34)(30,40,38,32);(2,6)(5,7)(10,16)(11,22)(12,14)(18,19)(20,36)(23,24)(25,39)(26,31)(27,28)(30,38)(32,34)(37,40);(1,2,3,7,4,6,8,5)(9,11,15,23)(10,17)(12,25,14,30)(13,24)(16,21,19,35)(18,33)(20,32)(22,29)(26,34)(27,39)(28,38)(31,40,36,37);(3,8)(5,7)(9,12)(10,18)(11,23)(13,28)(14,15)(17,32)(21,37)(27,29)(31,36)(33,34)(35,40)(38,39)Polycyclic presentation:
2^10|a^2=e,c^2=fj,f^2=ij,g^2=j,b^a=bej,c^a=cefij,c^b=cehi,d^a=dg,d^b=dj,d^c=dgi,e^c=eh,e^d=ej,f^a=fhi,f^b=fhj,f^d=fij,f^e=fj,g^b=gj,g^c=gi,g^d=gj,h^a=hj,h^c=hj61 conjugacy classes — view the full character table.
| maximal subgroup of | #72318794, #72318857, #72319038, … |
| maximal normal subgroup of | #72318794, #72318857, #72319038, … |
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.5);;
gap> SetPower(coll, 3, F.6*F.10);;
gap> SetPower(coll, 6, F.9*F.10);;
gap> SetPower(coll, 7, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.5*F.10);;
gap> SetConjugate(coll, 3, 1, F.3*F.5*F.6*F.9*F.10);;
gap> SetConjugate(coll, 3, 2, F.3*F.5*F.8*F.9);;
gap> SetConjugate(coll, 4, 1, F.4*F.7);;
gap> SetConjugate(coll, 4, 2, F.4*F.10);;
gap> SetConjugate(coll, 4, 3, F.4*F.7*F.9);;
gap> SetConjugate(coll, 5, 3, F.5*F.8);;
gap> SetConjugate(coll, 5, 4, F.5*F.10);;
gap> SetConjugate(coll, 6, 1, F.6*F.8*F.9);;
gap> SetConjugate(coll, 6, 2, F.6*F.8*F.10);;
gap> SetConjugate(coll, 6, 4, F.6*F.9*F.10);;
gap> SetConjugate(coll, 6, 5, F.6*F.10);;
gap> SetConjugate(coll, 7, 2, F.7*F.10);;
gap> SetConjugate(coll, 7, 3, F.7*F.9);;
gap> SetConjugate(coll, 7, 4, F.7*F.10);;
gap> SetConjugate(coll, 8, 1, F.8*F.10);;
gap> SetConjugate(coll, 8, 3, F.8*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);