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×#6869 |
| subgroup | G/H | |
|---|---|---|
| center | C3×C9 | H(3,𝔽3) |
| commutator | C32 | C32×C9 |
| Frattini, cosocle | C3×C9 | C33 |
| socle | C32 | C32⋊C9 |
| prime radical | C34 | C9 |
| LCS | G ▷ C32 ▷ C3 ▷ C1 | |
| UCS | C1 ◁ C3×C9 ◁ C32×C9 ◁ G | |
| LECS | G ▷ C3×C9 ▷ C3 ▷ C1 | |
208 conjugacy classes of subgroups.
| minimal degree | μ(G) = 30 |
| transitive degree | μt(G) = 81 |
A minimal faithful permutation representation — 3 generators on 30 points:
(1,2,3);(4,5,8,6,11,16,12,20,24,7,9,15,13,17,23,21,25,29,14,10,18,22,19,26,28,27,30);(5,9,10)(11,17,19)(20,25,27)153 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(729, 253);