Finite Group DatabasebetaFinite Group Databasebeta| order | 729 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of H(3,𝔽3) |
| automorphism | ? |
| homology | Abelianization (H1) = C33×C9 · Schur multiplier (H2) = C37 |
| direct product | C32×C32⋊C9 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C34 | C32 |
| commutator | C3 | C33×C9 |
| Frattini, cosocle | C32 | C34 |
| prime radical | C35 | C3 |
| LCS | G ▷ C3 ▷ C1 | |
| UCS | C1 ◁ C34 ◁ G | |
| LECS | G ▷ C32 ▷ C1 | |
1498 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 243 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2,3);(4,5,6);(7,8,10)(9,12,14)(11,15,13)(16,17,19,18,20,22,21,23,24);(8,11,12)(10,14,15)297 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(729, 416);