It could go either way. It doesn't necessarily need one, as the S doesn't stand for anything, but using a period would be more consistent compared to other presidents such as Franklin D. Roosevelt.
His parents chose "S" as his "middle name" in an attempt to please both of Harry's grandfathers, Anderson Shipp Truman and Solomon Young. The initial did not actually stand for anything, a common practice among the Scots-Irish.
Middle initials should always have a period after them, without exception.
Harry Truman
Harry S. Truman
Harry S Truman. His grandfathers, Anderson Shipp Truman and Solomon Young, could not agree on what Harry's middle name should be, and so Harry was given only a middle initial of "S", which stood for Solomon or Shipp, depending on which side of the family one asked.
Harry S Truman
Harry S. Truman
Harry S. Truman
Harry S. Truman was born on May 8, 1884.
He was born Harry S Truman
Harry Truman is not on money
Harry S Truman was US president at the end of World War 2 in 1945.Harry Truman
Harry S. Truman Harry S. Truman
Yes, if it's a real name. Some people have used a middle initial for no reason; one such well-known person was Harry Truman. His middle initial was "S", and was properly written without a period, thus: Harry S Truman. In such cases, a period should not be used.