Finite Group DatabasebetaFinite Group Databasebeta| order | 729 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of H(5,𝔽3) |
| automorphism | ? |
| homology | Abelianization (H1) = C35 · Schur multiplier (H2) = C39 |
| direct product | C3×H(5,𝔽3) |
| subgroup | G/H | |
|---|---|---|
| center, socle | C32 | C34 |
| commutator, Frattini, cosocle | C3 | C35 |
| LCS | G ▷ C3 ▷ C1 | |
| UCS | C1 ◁ C32 ◁ G | |
| LECS | G ▷ C3 ▷ C1 | |
3388 conjugacy classes of subgroups.
| minimal degree | μ(G) = 30 |
| transitive degree | μt(G) = 81 |
A minimal faithful permutation representation — 5 generators on 30 points:
(1,2,3);(4,5,8)(6,12,23)(7,11,18)(9,19,14)(10,22,16)(13,25,29)(15,21,17)(20,30,26)(24,28,27);(4,6,13)(5,9,20)(7,15,26)(8,17,28)(10,21,25)(11,12,24)(14,27,16)(18,19,29)(22,23,30);(6,14,15)(9,21,12)(13,26,27)(17,23,19)(20,24,25)(28,29,30);(5,10,11)(8,18,22)(9,21,12)(17,19,23)(20,25,24)(28,29,30)249 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(729, 501);