Finite Group DatabasebetaFinite Group Databasebeta| order | 729 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of H(3,𝔽9) |
| automorphism | ? |
| homology | Abelianization (H1) = C34 · Schur multiplier (H2) = C38 |
| unitriangular | UT(3,𝔽9) |
| Heisenberg | H(3,𝔽9) |
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C32 | C34 |
| LCS | G ▷ C32 ▷ C1 | |
| UCS | C1 ◁ C32 ◁ G | |
| LECS | G ▷ C32 ▷ C1 | |
667 conjugacy classes of subgroups.
| transitive degree | μt(G) = 81 |
Polycyclic presentation:
3^6|b^a=be,c^a=cf,d^a=def,c^b=ce^2f,d^b=de89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(729, 469);