Finite Group DatabasebetaFinite Group Databasebeta| order | 729 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6856 |
| automorphism | ? |
| homology | Abelianization (H1) = C33 · Schur multiplier (H2) = C33 |
| direct product | C3×#6859 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C33 | H(3,𝔽3) |
| commutator, Frattini, cosocle | C33 | C33 |
| prime radical | C32×H(3,𝔽3) | C3 |
| LCS | G ▷ C33 ▷ C32 ▷ C1 | |
| UCS | C1 ◁ C33 ◁ C34 ◁ G | |
| LECS | G ▷ C33 ▷ C32 ▷ C1 | |
244 conjugacy classes of subgroups.
| minimal degree | μ(G) = 39 |
| transitive degree | μt(G) = 243 |
A minimal faithful permutation representation — 3 generators on 39 points:
(1,2,3);(5,7,9)(13,14,17)(15,21,25)(16,20,26)(18,27,23)(19,30,24)(22,32,35)(28,38,39)(29,37,34)(31,36,33);(4,5,8)(6,9,12)(7,11,10)(13,15,22,24,34,39,16,23,33)(14,18,28,19,21,31,20,29,32)(17,25,35,30,37,38,26,27,36)105 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(729, 141);