Finite Group DatabasebetaFinite Group Databasebeta| order | 729 |
| nilpotency class | 1 |
| isoclinism | isoclinic class of C1 |
| automorphism | Aut(G) order 26244 |
| homology | Abelianization (H1) = G · Schur multiplier (H2) = C9 |
| abelian | C9×C81 |
| subgroup | G/H | |
|---|---|---|
| center | G | C1 |
| commutator | C1 | G |
| Frattini, cosocle | C3×C27 | C32 |
| socle, prime radical | C32 | C3×C27 |
| LCS | G ▷ C1 | |
| UCS | C1 ◁ G | |
| LECS | G ▷ C3×C27 ▷ C9 ▷ C3 ▷ C1 | |
49 conjugacy classes of subgroups.
| minimal degree | μ(G) = 90 |
| transitive degree | μt(G) = 729 |
A minimal faithful permutation representation — 2 generators on 90 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81);(82,83,84,85,86,87,88,89,90)729 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(729, 58);