Finite Group DatabasebetaFinite Group Databasebeta| order | 6561 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C37 · Schur multiplier (H2) = C320 |
| direct product | C3×3−1+6 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C32 | C36 |
| commutator, Frattini, cosocle | C3 | C37 |
| prime radical | C32×H(5,𝔽3) | C3 |
| LCS | G ▷ C3 ▷ C1 | |
| UCS | C1 ◁ C32 ◁ G | |
| LECS | G ▷ C3 ▷ C1 | |
2059386 conjugacy classes of subgroups.
| transitive degree | μt(G) = 243 |
2193 conjugacy classes. The character table is too large to display.
| maximal subgroup of | 3−1+8 |
| maximal normal subgroup of | 3−1+8 |
Past the SmallGroups library coverage — no SmallGroups id.