"taken to play at the big table in the Jockey Club at Buenos Aires"
Jockey Club, Buenos Aires (c.1910)
Public DomainJockey Club, Buenos Aires (c.1910) - Credit: unknown

The Argentinian Jockey Club, founded in 1882, was a body concerned with the control of horse-racing, but it also promoted social activities.

The Jockey Club at Buenos Aires, founded in 1887, is in the affluent Recoleta area of Buenos Aires, and is still operating.

The reference to the 'big table' may imply that the gambling game being played is the Punto banco version of the card game baccarat, sometimes known as 'big-table' baccarat.