Finite Group DatabasebetaFinite Group Databasebeta| order | 6400 |
| automorphism | ? |
| homology | Abelianization (H1) = C43 · Schur multiplier (H2) = C42 |
No aliases for this group.
| Sylow | C52 ⋊ C4×#347 |
965 conjugacy classes of subgroups; only the 208 normal subgroups are shown.
This group has 965 conjugacy classes of subgroups — too many to draw. Only its 208 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
Polycyclic presentation:
2^8,5^2|a^2=d,b^2=e,c^2=f,d^2=g,g^2=h,b^a=bgh,d^b=dh,e^a=eh,i^a=i^2,i^c=i^3,i^d=i^4,i^f=i^4,j^a=ij,j^c=j^3,j^d=i^3j,j^f=j^4148 conjugacy classes — view the full character table.
| maximal quotient of | #72242336 |
| holomorph of | #674 |
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, 5, 5]);;
gap> SetPower(coll, 1, F.4);;
gap> SetPower(coll, 2, F.5);;
gap> SetPower(coll, 3, F.6);;
gap> SetPower(coll, 4, F.7);;
gap> SetPower(coll, 7, F.8);;
gap> SetConjugate(coll, 2, 1, F.2*F.7*F.8);;
gap> SetConjugate(coll, 4, 2, F.4*F.8);;
gap> SetConjugate(coll, 5, 1, F.5*F.8);;
gap> SetConjugate(coll, 9, 1, F.9^2);;
gap> SetConjugate(coll, 9, 3, F.9^3);;
gap> SetConjugate(coll, 9, 4, F.9^4);;
gap> SetConjugate(coll, 9, 6, F.9^4);;
gap> SetConjugate(coll, 10, 1, F.9*F.10);;
gap> SetConjugate(coll, 10, 3, F.10^3);;
gap> SetConjugate(coll, 10, 4, F.9^3*F.10);;
gap> SetConjugate(coll, 10, 6, F.10^4);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);