Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C5 · Schur multiplier (H2) = C22 |
| unambiguous semidirect product | #63076⋊C5 |
| Fitting, Sylow | #63076 ⋊ C5 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C4 | C22×C24⋊C5 |
| commutator, hypocenter | 2−1+4 | C2×C4×C5 |
| Fitting | #63076 | C5 |
| socle | C2 | C2×C4×C24⋊C5 |
| prime radical | C2×#585⋊C5 | C2 |
| cosocle | #585 | C22×C5 |
| hypercenter | M16 | C24⋊C5 |
| LCS | G ▷ 2−1+4 | |
| UCS | C1 ◁ C4 ◁ M16 | |
1112 conjugacy classes of subgroups; only the 25 normal subgroups are shown.
This group has 1112 conjugacy classes of subgroups — too many to draw. Only its 25 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 128 |
| transitive degree | μt(G) = 128 |
Polycyclic presentation:
5,2^8|b^2=h,h^2=i,b^a=bdfghi,c^a=degh,d^a=cgi,e^a=defgh,f^a=cdg,g^b=gi,f^c=fi,e^d=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 1116340);