Finite Group DatabasebetaFinite Group Databasebeta| order | 729 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #10624930 |
| automorphism | ? |
| homology | Abelianization (H1) = C34 · Schur multiplier (H2) = C36 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C3 | C32×H(3,𝔽3) |
| commutator, Frattini, cosocle | C32 | C34 |
| prime radical | C3×H(3,𝔽3) | C32 |
| LCS | G ▷ C32 ▷ C3 ▷ C1 | |
| UCS | C1 ◁ C3 ◁ C3×H(3,𝔽3) ◁ G | |
| LECS | G ▷ C32 ▷ C3 ▷ C1 | |
468 conjugacy classes of subgroups.
| minimal degree | μ(G) = 81 |
| transitive degree | μt(G) = 81 |
Polycyclic presentation:
3^6|a^3=f,b^3=f^2,b^a=be,e^a=ef,d^c=df105 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(729, 493);