Finite Group DatabasebetaFinite Group Databasebeta| order | 729 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of C32⋊H(3,𝔽3) |
| automorphism | ? |
| homology | Abelianization (H1) = C34 · Schur multiplier (H2) = C35 |
| direct product | C3×#6895 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C33 | C33 |
| commutator, Frattini, cosocle | C32 | C34 |
| prime radical | C34 | C32 |
| LCS | G ▷ C32 ▷ C1 | |
| UCS | C1 ◁ C33 ◁ G | |
| LECS | G ▷ C32 ▷ C1 | |
418 conjugacy classes of subgroups.
| minimal degree | μ(G) = 39 |
| transitive degree | μt(G) = 243 |
A minimal faithful permutation representation — 4 generators on 39 points:
(1,2,3);(4,5,7,6,9,11,10,8,12)(13,14,17)(15,19,25)(16,20,24)(18,26,23)(21,28,33)(22,29,32)(27,34,31)(30,36,38)(35,39,37);(14,18,20)(17,24,26)(19,27,29)(25,32,34)(28,35,36)(33,38,39);(5,8,9)(7,11,12)(13,15,21,16,22,30,23,31,37)(14,19,28,20,29,36,18,27,35)(17,25,33,24,32,38,26,34,39)153 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(729, 431);