Finite Group DatabasebetaFinite Group Databasebeta| order | 729 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of H(3,𝔽3) |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C81 · Schur multiplier (H2) = C32 |
| unambiguous semidirect product | C32⋊C81 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C3×C27 | C32 |
| commutator | C3 | C3×C81 |
| socle | C32 | C3×C27 |
| prime radical | C33 | C27 |
| LCS | G ▷ C3 ▷ C1 | |
| UCS | C1 ◁ C3×C27 ◁ G | |
| LECS | G ▷ C3×C27 ▷ C9 ▷ C3 ▷ C1 | |
47 conjugacy classes of subgroups.
| transitive degree | μt(G) = 243 |
Polycyclic presentation:
3^6|a^3=d,d^3=e,e^3=f,b^a=bc297 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(729, 61);