From the Club's own website:

During the latter half of the 19th century, Buffalo was the hub of America's east-west commerce. Our railroad yards bustled with activity. The country's first grain elevator rose here on the shores of Lake Erie, and the metals industry had already established a firm foothold.

As commerce and industry flourished, they spurred cultural growth as well. The University of Buffalo, the Fine Arts Academy, the Buffalo Society of Natural Sciences and the Buffalo Historical Society were all founded during this prosperous era. It was during this same period that The Buffalo Club came into being.

Millard Fillmore, President of the United States from 1850 to 1853, had returned to Buffalo and was counted among the city's most influential citizens. In 1867, he joined 92 other like-minded gentlemen in signing The Buffalo Club's original incorporation papers. The Club's stated purpose was "to promote social intercourse among its members."

From 1867 to 1870, the Club leased the Movius home on the northwest corner of Delaware and Cary. When this lease expired, the Ganson house on the northwest corner of Delaware and Chippewa was purchased. In 1881 Grover Cleveland, then mayor of Buffalo, was elected to membership. Cleveland, who went on to serve two separate terms as America's President, remained a Buffalo Club member all his life. By 1887 the Club had outgrown the Ganson house, and the Pratt house was purchased. It remains the Club's home to this day. The roster of Buffalo Club members and special guests (both past and present) includes the names of many distinguished and influential personalities. It's been said that to know the Club's history is to know much of the history of Buffalo.

https://www.thebuffaloclub.org/about%20us

This is still the Buffalo Club, but the building has grown massively since 1905.