Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 5 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #136121 |
| commutator | C2×M32 | C22×C4 |
| Frattini, cosocle | C4×M32 | C23 |
| prime radical | C24 | #365 |
| LCS | G ▷ C2×M32 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C2×C4×C8 ◁ #33418 ◁ G | |
| LECS | G ▷ C4×M32 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
326 conjugacy classes of subgroups; only the 87 normal subgroups are shown.
This group has 326 conjugacy classes of subgroups — too many to draw. Only its 87 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 64 points:
(4,19)(8,30)(10,36)(13,43)(20,44)(23,57)(24,58)(32,35)(33,54)(34,63)(38,59)(39,45)(40,60)(41,62)(42,49)(46,64);(1,2)(3,13,14,45,16,43,18,39)(4,20,19,44)(5,25)(6,28)(7,9)(8,31,32,21,30,52,35,56)(10,38,36,59)(11,22)(12,29)(15,49,50,63,51,42,47,34)(17,54,48,64,53,33,26,46)(23,41,57,62)(24,60,58,40)(27,61)(37,55);(1,3,5,26,6,14,22,17,7,16,27,48,29,18,55,53)(2,8,11,42,12,32,25,34,9,30,37,49,28,35,61,63)(4,21,57,47,24,31,38,51,19,56,23,50,58,52,59,15)(10,39,60,64,41,43,20,54,36,45,40,46,62,13,44,33)Polycyclic presentation:
2^10|b^2=d,c^2=dgh,d^2=g,f^2=hi,h^2=i,i^2=j,b^a=bg,c^a=ce,c^b=cefhj,d^c=di,e^b=egj,f^a=fgj,f^b=fghij,f^c=fhij,f^d=fj,f^e=fj,g^c=gj,h^b=hi,h^c=hi,i^b=ij,i^c=ij64 conjugacy classes — view the full character table.
| maximal subgroup of | #72287678 |
| maximal normal subgroup of | #72287678 |
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.4);;
gap> SetPower(coll, 3, F.4*F.7*F.8);;
gap> SetPower(coll, 4, F.7);;
gap> SetPower(coll, 6, F.8*F.9);;
gap> SetPower(coll, 8, F.9);;
gap> SetPower(coll, 9, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.7);;
gap> SetConjugate(coll, 3, 1, F.3*F.5);;
gap> SetConjugate(coll, 3, 2, F.3*F.5*F.6*F.8*F.10);;
gap> SetConjugate(coll, 4, 3, F.4*F.9);;
gap> SetConjugate(coll, 5, 2, F.5*F.7*F.10);;
gap> SetConjugate(coll, 6, 1, F.6*F.7*F.10);;
gap> SetConjugate(coll, 6, 2, F.6*F.7*F.8*F.9*F.10);;
gap> SetConjugate(coll, 6, 3, F.6*F.8*F.9*F.10);;
gap> SetConjugate(coll, 6, 4, F.6*F.10);;
gap> SetConjugate(coll, 6, 5, F.6*F.10);;
gap> SetConjugate(coll, 7, 3, F.7*F.10);;
gap> SetConjugate(coll, 8, 2, F.8*F.9);;
gap> SetConjugate(coll, 8, 3, F.8*F.9);;
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);