Finite Group DatabasebetaFinite Group Databasebeta| order | 567 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of H(3,𝔽3) |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C7×C9 · Schur multiplier (H2) = C1 |
| direct product | C7×C27⋊C3 |
| unambiguous semidirect product | C27⋊(C3×C7) |
| subgroup | G/H | |
|---|---|---|
| center | C7×C9 | C32 |
| commutator | C3 | C3×C7×C9 |
| Frattini, cosocle | C9 | C32×C7 |
| socle | C3×C7 | C3×C9 |
| prime radical | C32×C7 | C9 |
| LCS | G ▷ C3 ▷ C1 | |
| UCS | C1 ◁ C7×C9 ◁ G | |
24 conjugacy classes of subgroups.
| minimal degree | μ(G) = 34 |
| transitive degree | μt(G) = 189 |
A minimal faithful permutation representation — 2 generators on 34 points:
(1,2,3,4,5,6,7)(8,9,12,10,14,20,16,23,28,11,15,19,17,24,27,25,31,33,18,13,21,26,22,29,32,30,34);(9,13,15)(12,19,21)(14,22,24)(20,27,29)(23,30,31)(28,33,34)231 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(567, 28);