Finite Group DatabasebetaFinite Group Databasebeta| order | 729 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of H(3,𝔽3) |
| automorphism | ? |
| homology | Abelianization (H1) = C35 · Schur multiplier (H2) = C39 |
| direct product | C33×3−1+2 |
| unambiguous semidirect product | C9⋊C34 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C34 | C32 |
| commutator, Frattini, cosocle | C3 | C35 |
| prime radical | C35 | C3 |
| LCS | G ▷ C3 ▷ C1 | |
| UCS | C1 ◁ C34 ◁ G | |
| LECS | G ▷ C3 ▷ C1 | |
3604 conjugacy classes of subgroups.
| minimal degree | μ(G) = 18 |
| transitive degree | μt(G) = 243 |
A minimal faithful permutation representation — 5 generators on 18 points:
(1,2,3);(4,5,6);(7,8,9);(10,11,13,12,15,17,16,14,18);(11,14,15)(13,17,18)297 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(729, 499);