Finite Group DatabasebetaFinite Group Databasebeta| order | 6561 |
| nilpotency class | 1 |
| isoclinism | isoclinic class of C1 |
| automorphism | Aut(G) order 26244 |
| homology | Abelianization (H1) = G · Schur multiplier (H2) = C3 |
| abelian | C3×C2187 |
| subgroup | G/H | |
|---|---|---|
| center | G | C1 |
| commutator | C1 | G |
| Frattini, cosocle | C729 | C32 |
| socle, prime radical | C32 | C729 |
| LCS | G ▷ C1 | |
| UCS | C1 ◁ G | |
| LECS | G ▷ C729 ▷ C243 ▷ C81 ▷ C27 ▷ C9 ▷ C3 ▷ C1 | |
30 conjugacy classes of subgroups.
6561 conjugacy classes. The character table is too large to display.
| maximal subgroup of | C2×C3×C2187, C3×C6561, C9×C2187, … |
| maximal normal subgroup of | C2×C3×C2187, C3×C6561, C9×C2187, … |
| maximal quotient of | C2×C3×C2187, C3×C6561, C9×C2187, … |
Past the SmallGroups library coverage — no SmallGroups id.