Finite Group DatabasebetaFinite Group Databasebeta| order | 2560 |
| automorphism | ? |
| homology | Abelianization (H1) = C2 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Fitting | #17309 ⋊ D5 |
| derived | #17309⋊C5 ⋊ C2 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C1 | G |
| commutator, cosocle, hypocenter | #17309⋊C5 | C2 |
| Frattini, socle | C24 | (C24⋊C5)⋊C2 |
| Fitting | #17309 | D5 |
| LCS | G ▷ #17309⋊C5 | |
| UCS | C1 | |
329 conjugacy classes of subgroups; only the 5 normal subgroups are shown.
This group has 329 conjugacy classes of subgroups — too many to draw. Only its 5 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|a^2=j,c^2=i,d^2=gh,e^2=hi,f^2=hj,b^a=b^4hi,c^b=cd,d^a=cfgi,d^b=ceghij,d^c=dg,e^a=cegij,e^b=cfghi,e^c=ej,e^d=ehij,f^a=cdgi,f^b=cghi,f^c=fgi,f^d=fgij,f^e=fghij,g^a=gi,g^b=ghi,h^a=hij,h^b=gij,i^b=hi,j^b=gj22 conjugacy classes — view the full character table.
| maximal subgroup of | #72209709, C2×#72209706, #72235032 |
| maximal normal subgroup of | C2×#72209706, #72235032 |
| maximal quotient of | #72210082, C2×#72209706, #72266660, … |
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> SetPower(coll, 1, F.10);;
gap> SetPower(coll, 3, F.9);;
gap> SetPower(coll, 4, F.7*F.8);;
gap> SetPower(coll, 5, F.8*F.9);;
gap> SetPower(coll, 6, F.8*F.10);;
gap> SetConjugate(coll, 2, 1, F.2^4*F.8*F.9);;
gap> SetConjugate(coll, 3, 2, F.3*F.4);;
gap> SetConjugate(coll, 4, 1, F.3*F.6*F.7*F.9);;
gap> SetConjugate(coll, 4, 2, F.3*F.5*F.7*F.8*F.9*F.10);;
gap> SetConjugate(coll, 4, 3, F.4*F.7);;
gap> SetConjugate(coll, 5, 1, F.3*F.5*F.7*F.9*F.10);;
gap> SetConjugate(coll, 5, 2, F.3*F.6*F.7*F.8*F.9);;
gap> SetConjugate(coll, 5, 3, F.5*F.10);;
gap> SetConjugate(coll, 5, 4, F.5*F.8*F.9*F.10);;
gap> SetConjugate(coll, 6, 1, F.3*F.4*F.7*F.9);;
gap> SetConjugate(coll, 6, 2, F.3*F.7*F.8*F.9);;
gap> SetConjugate(coll, 6, 3, F.6*F.7*F.9);;
gap> SetConjugate(coll, 6, 4, F.6*F.7*F.9*F.10);;
gap> SetConjugate(coll, 6, 5, F.6*F.7*F.8*F.9*F.10);;
gap> SetConjugate(coll, 7, 1, F.7*F.9);;
gap> SetConjugate(coll, 7, 2, F.7*F.8*F.9);;
gap> SetConjugate(coll, 8, 1, F.8*F.9*F.10);;
gap> SetConjugate(coll, 8, 2, F.7*F.9*F.10);;
gap> SetConjugate(coll, 9, 2, F.8*F.9);;
gap> SetConjugate(coll, 10, 2, F.7*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);