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 | C3×#6872 |
| subgroup | G/H | |
|---|---|---|
| center | C3×C9 | H(3,𝔽3) |
| commutator | C32 | C32×C9 |
| Frattini, cosocle | C3×C9 | C33 |
| socle | C32 | C32⋊C9 |
| prime radical | C3×H(3,𝔽3) | C9 |
| LCS | G ▷ C32 ▷ C3 ▷ C1 | |
| UCS | C1 ◁ C3×C9 ◁ C32×C9 ◁ G | |
| LECS | G ▷ C3×C9 ▷ C3 ▷ C1 | |
160 conjugacy classes of subgroups.
| minimal degree | μ(G) = 84 |
| transitive degree | μt(G) = 243 |
A minimal faithful permutation representation — 3 generators on 84 points:
(1,2,3);(4,5,9,7,12,24,19,30,46,8,13,23,20,31,45,41,55,68,21,11,27,42,29,51,65,54,72)(6,14,32,17,34,57,39,59,76,18,28,50,40,53,71,64,74,82,16,10,22,38,26,44,63,49,67)(15,33,43,36,58,66,61,77,80,35,52,56,60,73,75,78,83,84,37,25,47,62,48,69,79,70,81);(4,6,15)(5,10,25)(7,17,36)(8,18,35)(9,22,43)(11,28,52)(12,26,48)(13,14,33)(16,37,21)(19,39,61)(20,40,60)(23,32,56)(24,44,66)(27,50,47)(29,53,73)(30,49,70)(31,34,58)(38,62,42)(41,64,78)(45,57,75)(46,67,80)(51,71,69)(54,74,83)(55,59,77)(63,79,65)(68,76,84)(72,82,81)153 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(729, 261);