The definition of “minority” is “the smaller of two groups forming a whole; a group of things or persons numbering less than half of a total.” In other words, take any group of things or persons and divide it into two groups. The larger group is the “majority” and the smaller one is the “minority.’’
The primary meaning of the word ”minority” does not indicate any specific group of people or things. It refers only to size.
The secondary meaning of the word refers to a group regarded as different from the larger group of which it is a part.
Without the context, your example, gives no indication to what it refers, people or things.
“Largish” means fairly large. So a “largish minority” would be group of persons or things that is relatively close to half of the whole.
BTW, if you can get your hands on “The American Heritage Dictionary of the English Language,” you would find, not only good definitions of words, but also examples of usage. Unfortunately, I think it’s pretty expensive in China. Nevertheless, it’s a tool I continue to use on a regular basis.