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) = C34 |
| direct product | C3×#6866 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C33 | H(3,𝔽3) |
| commutator | C32 | C32×C9 |
| Frattini, cosocle | C33 | C33 |
| prime radical | C35 | C3 |
| LCS | G ▷ C32 ▷ C3 ▷ C1 | |
| UCS | C1 ◁ C33 ◁ C34 ◁ G | |
| LECS | G ▷ C33 ▷ C3 ▷ C1 | |
994 conjugacy classes of subgroups.
| minimal degree | μ(G) = 21 |
| transitive degree | μt(G) = 81 |
A minimal faithful permutation representation — 3 generators on 21 points:
(1,2,3);(4,5,7,6,8,10,9,11,12)(13,14,16)(15,17,20)(18,21,19);(14,17,18)153 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(729, 245);