Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C5 · Schur multiplier (H2) = C24 |
| direct product | C2×H(7,𝔽2)⋊C5 |
| unambiguous semidirect product | (C2×H(7,𝔽2))⋊C5 |
| Fitting, Sylow | C2×H(7,𝔽2) ⋊ C5 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×C24⋊C5 |
| commutator, cosocle, hypocenter | 2−1+4 | C23×C5 |
| Frattini | C2 | C23×C24⋊C5 |
| Fitting | C2×H(7,𝔽2) | C5 |
| hypercenter | C2×Q8 | C24⋊C5 |
| LCS | G ▷ 2−1+4 | |
| UCS | C1 ◁ C22 ◁ C2×Q8 | |
6120 conjugacy classes of subgroups; only the 51 normal subgroups are shown.
This group has 6120 conjugacy classes of subgroups — too many to draw. Only its 51 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 160 |
74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 1116353);