Finite Group DatabasebetaFinite Group Databasebeta| order | 729 |
| nilpotency class | 1 |
| isoclinism | isoclinic class of C1 |
| automorphism | Aut(G) order 147386304 |
| homology | Abelianization (H1) = G · Schur multiplier (H2) = C36 |
| abelian | C33×C27 |
| subgroup | G/H | |
|---|---|---|
| center | G | C1 |
| commutator | C1 | G |
| Frattini, cosocle | C9 | C34 |
| socle, prime radical | C34 | C9 |
| LCS | G ▷ C1 | |
| UCS | C1 ◁ G | |
| LECS | G ▷ C9 ▷ C3 ▷ C1 | |
580 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 729 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27);(28,29,30);(31,32,33);(34,35,36)729 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(729, 475);