Finite Group DatabasebetaFinite Group Databasebeta| order | 864 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
| direct product | #41×D27 |
| Fitting | #41×C27 ⋊ C2 |
| Sylow | C27 ⋊ C2×#41 |
| subgroup | G/H | |
|---|---|---|
| center | C4 | C22×D27 |
| commutator, cosocle | C2×C27 | C24 |
| Frattini | C2×C9 | C23×D3 |
| Fitting | #41×C27 | C2 |
| socle | C2×C3 | C23×D9 |
| hypercenter | #41 | D27 |
| hypocenter | C27 | C2×#41 |
| LCS | G ▷ C2×C27 ▷ C27 | |
| UCS | C1 ◁ C4 ◁ #41 | |
328 conjugacy classes of subgroups; only the 121 normal subgroups are shown.
This group has 328 conjugacy classes of subgroups — too many to draw. Only its 121 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 216 |
150 conjugacy classes — view the full character table.
| maximal subgroup of | #104×D27, #13134326, #131×D27, … |
| maximal normal subgroup of | #104×D27, #13134326, #131×D27, … |
| maximal quotient of | #13136255, #13136256, #13136257, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(864, 641);