Finite Group DatabasebetaFinite Group Databasebeta| order | 2560 |
| automorphism | ? |
| homology | Abelianization (H1) = C2 · Schur multiplier (H2) = C26 |
No aliases for this group.
| Fitting, socle | C28 ⋊ D5 |
| derived | C24⋊(C24⋊C5) ⋊ C2 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, cosocle, hypocenter | C24⋊(C24⋊C5) | C2 |
| Fitting, socle | C28 | D5 |
| LCS | G ▷ C24⋊(C24⋊C5) | |
| UCS | C1 | |
46657 conjugacy classes of subgroups; only the 9 normal subgroups are shown.
This group has 46657 conjugacy classes of subgroups — too many to draw. Only its 9 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
Polycyclic presentation:
2,5,2^8|b^a=b^4,c^a=cf,c^b=d,d^a=cef,d^b=cde,e^a=cd,e^b=f,f^b=cd,g^a=ghi,g^b=gh,h^a=j,h^b=ij,i^a=hij,i^b=gi,j^a=h,j^b=gij52 conjugacy classes — view the full character table.
| maximal subgroup of | #72209703, #72209715, #72209721, … |
| maximal normal subgroup of | #72218553, #72218667, C2×#72209701, … |
| maximal quotient of | #72210032, #72210041, #72211305, … |
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, 5, 2, 2, 2, 2, 2, 2, 2, 2]);;
gap> SetConjugate(coll, 2, 1, F.2^4);;
gap> SetConjugate(coll, 3, 1, F.3*F.6);;
gap> SetConjugate(coll, 3, 2, F.4);;
gap> SetConjugate(coll, 4, 1, F.3*F.5*F.6);;
gap> SetConjugate(coll, 4, 2, F.3*F.4*F.5);;
gap> SetConjugate(coll, 5, 1, F.3*F.4);;
gap> SetConjugate(coll, 5, 2, F.6);;
gap> SetConjugate(coll, 6, 2, F.3*F.4);;
gap> SetConjugate(coll, 7, 1, F.7*F.8*F.9);;
gap> SetConjugate(coll, 7, 2, F.7*F.8);;
gap> SetConjugate(coll, 8, 1, F.10);;
gap> SetConjugate(coll, 8, 2, F.9*F.10);;
gap> SetConjugate(coll, 9, 1, F.8*F.9*F.10);;
gap> SetConjugate(coll, 9, 2, F.7*F.9);;
gap> SetConjugate(coll, 10, 1, F.8);;
gap> SetConjugate(coll, 10, 2, F.7*F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);