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 | C32×#737 |
| subgroup | G/H | |
|---|---|---|
| center | C32×C9 | C32 |
| commutator, Frattini, cosocle | C3 | C35 |
| socle | C33 | C33 |
| prime radical | C32×H(3,𝔽3) | C3 |
| LCS | G ▷ C3 ▷ C1 | |
| UCS | C1 ◁ C32×C9 ◁ G | |
| LECS | G ▷ C3 ▷ C1 | |
2902 conjugacy classes of subgroups.
| minimal degree | μ(G) = 33 |
| transitive degree | μt(G) = 243 |
A minimal faithful permutation representation — 5 generators on 33 points:
(1,2,3);(4,5,6);(7,8,11)(9,13,19)(10,14,18)(12,20,17)(15,22,27)(16,23,26)(21,28,25)(24,30,32)(29,33,31);(8,12,14)(11,18,20)(13,21,23)(19,26,28)(22,29,30)(27,32,33);(7,9,15,10,16,24,17,25,31)(8,13,22,14,23,30,12,21,29)(11,19,27,18,26,32,20,28,33)297 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(729, 500);