Finite Group DatabasebetaFinite Group Databasebeta| order | 729 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of H(3,ℤ/9) |
| automorphism | ? |
| homology | Abelianization (H1) = C92 · Schur multiplier (H2) = C1 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator | C9 | C92 |
| Frattini, cosocle | C3×C27 | C32 |
| socle | C3 | #6864 |
| prime radical | C32 | C27⋊C3 |
| LCS | G ▷ C9 ▷ C1 | |
| UCS | C1 ◁ C9 ◁ G | |
| LECS | G ▷ C3×C27 ▷ C9 ▷ C3 ▷ C1 | |
33 conjugacy classes of subgroups.
| minimal degree | μ(G) = 81 |
| transitive degree | μt(G) = 81 |
Polycyclic presentation:
3^6|a^3=c,b^3=d,c^3=e,e^3=f,b^a=be,d^a=df,c^b=cf^2105 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(729, 60);