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 |
| direct product | C9×#731 |
| 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 | |
146 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 243 |
A minimal faithful permutation representation — 3 generators on 36 points:
(1,2,3,4,5,6,7,8,9);(10,11,14)(12,18,22)(13,17,23)(15,24,20)(16,27,21)(19,29,32)(25,35,36)(26,34,31)(28,33,30);(10,12,19,13,20,30,21,31,36)(11,15,25,17,26,29,16,18,28)(14,22,32,23,24,33,27,34,35)153 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(729, 277);