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 | C32×#733 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C33 | H(3,𝔽3) |
| commutator, Frattini, cosocle | C32 | C34 |
| prime radical | C34 | C32 |
| LCS | G ▷ C32 ▷ C3 ▷ C1 | |
| UCS | C1 ◁ C33 ◁ C34 ◁ G | |
| LECS | G ▷ C32 ▷ C3 ▷ C1 | |
526 conjugacy classes of subgroups.
| minimal degree | μ(G) = 33 |
| transitive degree | μt(G) = 243 |
A minimal faithful permutation representation — 4 generators on 33 points:
(1,2,3);(4,5,6);(7,8,11,10,14,20,18,13,24)(9,15,19,17,12,21,28,23,31)(16,26,29,27,25,30,33,22,32);(7,9,16,18,28,33,10,17,27)(8,12,22,13,15,25,14,23,26)(11,19,29,24,31,32,20,21,30)153 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(729, 482);