Finite Group DatabasebetaFinite Group Databasebeta| order | 5120 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C5 · Schur multiplier (H2) = C27 |
| direct product | C2×#72210028 |
| Fitting, Sylow | C2×H(9,𝔽2) ⋊ C5 |
| derived | H(9,𝔽2) ⋊ C2×C5 |
| subgroup | G/H | |
|---|---|---|
| center, socle, hypercenter | C22 | C24⋊(C24⋊C5) |
| commutator, cosocle, hypocenter | H(9,𝔽2) | C2×C5 |
| Frattini | C2 | C2×C24⋊(C24⋊C5) |
| Fitting | C2×H(9,𝔽2) | C5 |
| LCS | G ▷ H(9,𝔽2) | |
| UCS | C1 ◁ C22 | |
At least 8252 conjugacy classes of subgroups; only the 43 normal subgroups are shown.
This group has at least 8252 conjugacy classes of subgroups — too many to draw. Only its 43 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
122 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#72210032, #72256667, C2×#72210041, … |
| maximal normal subgroup of | C2×#72210032, #72256667, C2×#72210041, … |
Past the SmallGroups library coverage — no SmallGroups id.