Finite Group DatabasebetaFinite Group Databasebeta| order | 729 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of H(3,𝔽3) |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C92 · Schur multiplier (H2) = C33 |
| direct product | C3×#6864 |
| subgroup | G/H | |
|---|---|---|
| center | C32×C9 | C32 |
| commutator | C3 | C3×C92 |
| Frattini, cosocle | C3×C9 | C33 |
| socle, prime radical | C33 | C3×C9 |
| LCS | G ▷ C3 ▷ C1 | |
| UCS | C1 ◁ C32×C9 ◁ G | |
| LECS | G ▷ C3×C9 ▷ C3 ▷ C1 | |
166 conjugacy classes of subgroups.
| minimal degree | μ(G) = 39 |
| transitive degree | μt(G) = 729 |
A minimal faithful permutation representation — 3 generators on 39 points:
(1,2,3);(13,14,17,15,19,25,21,28,33,16,20,24,22,29,32,30,36,38,23,18,26,31,27,34,37,35,39);(4,5,7,6,8,10,9,11,12)(14,18,20)(17,24,26)(19,27,29)(25,32,34)(28,35,36)(33,38,39)297 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(729, 239);