Finite Group DatabasebetaFinite Group Databasebeta| order | 729 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of AΣL(1,𝔽27) |
| automorphism | ? |
| homology | Abelianization (H1) = C32×C9 · Schur multiplier (H2) = C33 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C3×C9 | H(3,𝔽3) |
| commutator | C32 | C32×C9 |
| Frattini, cosocle | C33 | C33 |
| socle | C32 | C3×H(3,𝔽3) |
| prime radical | C3×H(3,𝔽3) | C32 |
| LCS | G ▷ C32 ▷ C3 ▷ C1 | |
| UCS | C1 ◁ C3×C9 ◁ C32×C9 ◁ G | |
| LECS | G ▷ C33 ▷ C3 ▷ C1 | |
184 conjugacy classes of subgroups.
| transitive degree | μt(G) = 243 |
Polycyclic presentation:
3^6|a^3=e,c^3=f,b^a=bd,d^a=df,d^b=df153 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(729, 257);