Finite Group DatabasebetaFinite Group Databasebeta| order | 729 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of AΣL(1,𝔽27) |
| automorphism | ? |
| homology | Abelianization (H1) = C34 · Schur multiplier (H2) = C36 |
| direct product | C3×#6908 |
| subgroup | G/H | |
|---|---|---|
| center | C3×C9 | H(3,𝔽3) |
| commutator, Frattini, cosocle | C32 | C34 |
| socle | C32 | C3×H(3,𝔽3) |
| LCS | G ▷ C32 ▷ C3 ▷ C1 | |
| UCS | C1 ◁ C3×C9 ◁ C32×C9 ◁ G | |
| LECS | G ▷ C32 ▷ C3 ▷ C1 | |
532 conjugacy classes of subgroups.
| minimal degree | μ(G) = 30 |
| transitive degree | μt(G) = 81 |
A minimal faithful permutation representation — 4 generators on 30 points:
(1,2,3);(4,5,8)(6,10,15)(7,9,16)(11,20,14)(12,18,23)(13,17,24)(19,27,22)(21,25,29)(26,30,28);(5,9,11)(10,17,19)(18,25,26);(4,6,12,7,13,21,14,22,28)(5,10,18,9,17,25,11,19,26)(8,15,23,16,24,29,20,27,30)153 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(729, 483);