Finite Group DatabasebetaFinite Group Databasebeta| order | 6400 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C8 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Sylow | C52 ⋊ C2×#1313 |
852 conjugacy classes of subgroups; only the 187 normal subgroups are shown.
This group has 852 conjugacy classes of subgroups — too many to draw. Only its 187 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 640 |
Polycyclic presentation:
2^8,5^2|a^2=d,b^2=e,d^2=f,e^2=g,g^2=h,b^a=bg,d^b=dh,e^a=eh,i^a=i^3,i^b=i^4,i^c=i^4,i^d=i^4,j^a=j^3,j^b=i^4j,j^c=j^4,j^d=j^4232 conjugacy classes — view the full character table.
| maximal subgroup of | #72242215, #72242317 |
| maximal normal subgroup of | #72242215, #72242317 |
| maximal quotient of | #72269190 |
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, 4, F.6);;
gap> SetPower(coll, 5, F.7);;
gap> SetPower(coll, 7, F.8);;
gap> SetConjugate(coll, 2, 1, F.2*F.7);;
gap> SetConjugate(coll, 4, 2, F.4*F.8);;
gap> SetConjugate(coll, 5, 1, F.5*F.8);;
gap> SetConjugate(coll, 9, 1, F.9^3);;
gap> SetConjugate(coll, 9, 2, F.9^4);;
gap> SetConjugate(coll, 9, 3, F.9^4);;
gap> SetConjugate(coll, 9, 4, F.9^4);;
gap> SetConjugate(coll, 10, 1, F.10^3);;
gap> SetConjugate(coll, 10, 2, F.9^4*F.10);;
gap> SetConjugate(coll, 10, 3, F.10^4);;
gap> SetConjugate(coll, 10, 4, F.10^4);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);