Finite Group DatabasebetaFinite Group Databasebeta| order | 729 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of H(3,𝔽3) |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C92 · Schur multiplier (H2) = C33×C9 |
| direct product | C9×C32⋊C9 |
| unambiguous semidirect product | C32⋊C92 |
| subgroup | G/H | |
|---|---|---|
| center | C32×C9 | C32 |
| commutator | C3 | C3×C92 |
| Frattini, socle, cosocle | C33 | C33 |
| prime radical | C34 | C32 |
| LCS | G ▷ C3 ▷ C1 | |
| UCS | C1 ◁ C32×C9 ◁ G | |
| LECS | G ▷ C33 ▷ C1 | |
331 conjugacy classes of subgroups.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 243 |
A minimal faithful permutation representation — 3 generators on 27 points:
(1,2,3,4,5,6,7,8,9);(10,11,13)(12,15,17)(14,18,16)(19,20,22,21,23,25,24,26,27);(11,14,15)(13,17,18)297 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(729, 105);