Finite Group DatabasebetaFinite Group Databasebeta| order | 729 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6909 |
| automorphism | ? |
| homology | Abelianization (H1) = C34 · Schur multiplier (H2) = C36 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C9 | C3×H(3,𝔽3) |
| commutator, Frattini, cosocle | C32 | C34 |
| socle | C3 | C32×H(3,𝔽3) |
| LCS | G ▷ C32 ▷ C3 ▷ C1 | |
| UCS | C1 ◁ C9 ◁ C32×C9 ◁ G | |
| LECS | G ▷ C32 ▷ C3 ▷ C1 | |
480 conjugacy classes of subgroups.
| minimal degree | μ(G) = 81 |
| transitive degree | μt(G) = 81 |
Polycyclic presentation:
3^6|d^3=f,b^a=be,e^a=ef,c^b=cf105 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(729, 489);