Finite Group DatabasebetaFinite Group Databasebeta| order | 729 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6909 |
| automorphism | ? |
| homology | Abelianization (H1) = C34 · Schur multiplier (H2) = C37 |
| direct product | C3×#6911 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C32 | C3×H(3,𝔽3) |
| commutator, Frattini, cosocle | C32 | C34 |
| LCS | G ▷ C32 ▷ C3 ▷ C1 | |
| UCS | C1 ◁ C32 ◁ C34 ◁ G | |
| LECS | G ▷ C32 ▷ C3 ▷ C1 | |
652 conjugacy classes of subgroups.
| minimal degree | μ(G) = 30 |
| transitive degree | μt(G) = 81 |
A minimal faithful permutation representation — 4 generators on 30 points:
(1,2,3);(4,5,8)(6,12,17)(7,11,18)(9,19,15)(10,22,16)(13,24,27)(14,21,28)(20,30,26)(23,29,25);(4,6,13,16,14,26,7,15,25)(5,9,20,10,12,23,11,21,24)(8,17,27,22,28,30,18,19,29);(6,14,15)(9,12,21)(13,25,26)(17,28,19)(20,24,23)(27,29,30)105 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(729, 486);