Finite Group DatabasebetaFinite Group Databasebeta| order | 8000 |
| automorphism | ? |
| homology | Abelianization (H1) = C42×C5 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Fitting | M16×5−1+2 ⋊ C4 |
| Sylow | 5−1+2 ⋊ #339 |
340 conjugacy classes of subgroups; only the 79 normal subgroups are shown.
This group has 340 conjugacy classes of subgroups — too many to draw. Only its 79 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 200 |
Polycyclic presentation:
2^2,5,2^4,5^2|a^2=d,b^2=de,f^2=g,h^5=i,b^a=bf,e^a=eg,f^a=fg,h^a=h^3i^3,h^b=h^3i^3,h^c=hi^3,h^d=h^4i^4,i^a=i^3,i^b=i^3,i^d=i^4200 conjugacy classes — view the full character table.
| maximal subgroup of | #104×AGL(1,ℤ/25) |
| maximal normal subgroup of | #104×AGL(1,ℤ/25) |
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");;
gap> coll := SingleCollector(F, [2, 2, 5, 2, 2, 2, 2, 5, 5]);;
gap> SetPower(coll, 1, F.4);;
gap> SetPower(coll, 2, F.4*F.5);;
gap> SetPower(coll, 6, F.7);;
gap> SetPower(coll, 8, F.9);;
gap> SetConjugate(coll, 2, 1, F.2*F.6);;
gap> SetConjugate(coll, 5, 1, F.5*F.7);;
gap> SetConjugate(coll, 6, 1, F.6*F.7);;
gap> SetConjugate(coll, 8, 1, F.8^3*F.9^3);;
gap> SetConjugate(coll, 8, 2, F.8^3*F.9^3);;
gap> SetConjugate(coll, 8, 3, F.8*F.9^3);;
gap> SetConjugate(coll, 8, 4, F.8^4*F.9^4);;
gap> SetConjugate(coll, 9, 1, F.9^3);;
gap> SetConjugate(coll, 9, 2, F.9^3);;
gap> SetConjugate(coll, 9, 4, F.9^4);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);