Finite Group DatabasebetaFinite Group Databasebeta| order | 729 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #10624505 |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C9 · Schur multiplier (H2) = C33 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C3 | #6866 |
| commutator | C33 | C3×C9 |
| Frattini, cosocle | C34 | C32 |
| prime radical | C32×H(3,𝔽3) | C3 |
| LCS | G ▷ C33 ▷ C32 ▷ C3 ▷ C1 | |
| UCS | C1 ◁ C3 ◁ C33 ◁ C34 ◁ G | |
| LECS | G ▷ C34 ▷ C32 ▷ C3 ▷ C1 | |
175 conjugacy classes of subgroups.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 27 |
Polycyclic presentation:
3^6|a^3=d,b^a=bc,c^a=ce,e^a=ef,d^b=df^257 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(729, 65);