Finite Group DatabasebetaFinite Group Databasebeta| order | 3200 |
| automorphism | ? |
| homology | Abelianization (H1) = C4×C8 · Schur multiplier (H2) = C1 |
No aliases for this group.
| Sylow | C52 ⋊ #1398 |
142 conjugacy classes of subgroups; only the 69 normal subgroups are shown.
This group has 142 conjugacy classes of subgroups — too many to draw. Only its 69 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 160 |
Polycyclic presentation:
2^7,5^2|a^2=c,b^2=d,c^2=e,e^2=f,f^2=g,b^a=bf,c^b=cg,d^a=dg,h^a=h^3,h^c=h^4,i^a=h^2i^4,i^c=h^4i284 conjugacy classes — view the full character table.
| maximal subgroup of | #72242233, #72242280, #72242282, … |
| maximal normal subgroup of | #72242233, #72242280, #72242282, … |
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, 2, 2, 2, 2, 2, 5, 5]);;
gap> SetPower(coll, 1, F.3);;
gap> SetPower(coll, 2, F.4);;
gap> SetPower(coll, 3, F.5);;
gap> SetPower(coll, 5, F.6);;
gap> SetPower(coll, 6, F.7);;
gap> SetConjugate(coll, 2, 1, F.2*F.6);;
gap> SetConjugate(coll, 3, 2, F.3*F.7);;
gap> SetConjugate(coll, 4, 1, F.4*F.7);;
gap> SetConjugate(coll, 8, 1, F.8^3);;
gap> SetConjugate(coll, 8, 3, F.8^4);;
gap> SetConjugate(coll, 9, 1, F.8^2*F.9^4);;
gap> SetConjugate(coll, 9, 3, F.8^4*F.9);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);