Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25×C4 |
| direct product | #127×D27 |
| Fitting | C27×#127 ⋊ C2 |
| Sylow | C27 ⋊ C2×#127 |
| subgroup | G/H | |
|---|---|---|
| center | C22 | C23×D27 |
| commutator, cosocle | C22×C27 | C24 |
| Frattini | C22×C9 | C23×D3 |
| Fitting | C27×#127 | C2 |
| socle | C22×C3 | C23×D9 |
| hypercenter | #127 | D27 |
| hypocenter | C27 | C2×#127 |
| LCS | G ▷ C22×C27 ▷ C27 | |
| UCS | C1 ◁ C22 ◁ #127 | |
996 conjugacy classes of subgroups; only the 183 normal subgroups are shown.
This group has 996 conjugacy classes of subgroups — too many to draw. Only its 183 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 432 |
210 conjugacy classes — view the full character table.
| maximal subgroup of | #127×D27⋊C3 |
| maximal normal subgroup of | #127×D27⋊C3 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 2516);