Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| automorphism | ? |
| homology | Abelianization (H1) = C5×C8 · Schur multiplier (H2) = C2 |
| unambiguous semidirect product | #62624⋊C5 |
| Fitting, Sylow | #62624 ⋊ C5 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C16 | C24⋊C5 |
| commutator, hypocenter | 2−1+4 | C5×C8 |
| Frattini | C8 | C2×C24⋊C5 |
| Fitting | #62624 | C5 |
| socle | C2 | C8×C24⋊C5 |
| prime radical | #585⋊C5 | C4 |
| cosocle | #3573 | C2×C5 |
| LCS | G ▷ 2−1+4 | |
| UCS | C1 ◁ C16 | |
220 conjugacy classes of subgroups; only the 13 normal subgroups are shown.
This group has 220 conjugacy classes of subgroups — too many to draw. Only its 13 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 256 |
Polycyclic presentation:
5,2^8|b^2=g,g^2=h,h^2=i,c^a=dfi,d^a=cdefi,e^a=fi,f^a=deh,f^c=fi,e^d=ei104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 1116317);