Fascism is a political philosophy that glorifies the state, or the collective, as opposed to the individual. Fascists believe that government should be centralized and have one strong leader (a dictator, usually). Fascists believe that the nation functions better as a unit and will do anything to achieve such control. In order to achieve this control, fascists suppress their opposition through terror, censorship, rabid nationalism and sometimes racism.
When you compare this ideology to Miss Brodie, can you see how her character could be fascist in nature? She gathers about her a group of girls. She expects them to function as a unit. She expects them to espose her beliefs. The Brodie Set is a unit and functions as a unit. When some members try to break way, Miss Brodie exerts control and tries to prevent this through intimidation and bullying (as with Mary MacGregor). Miss Brodie is dictatorial in how she manipulates "her girls". She, however, can behave any way she wants, and her behavior is immoral and inappropriate for a teacher who is supposed to be a role model. Like a dictator, her actions are above reproach. She ignores laws when it suits her fancy, for the better good. For example, she ignores the curriculum at Marcia Blaine School for Girls because she believes her methods of teaching and the subject matter she chooses is superior. The reason she admires fascism is because of its order, and she is orderly to a fault. She is obsessive about it. Like a fascist, however, she ignores the negative effects that this philosophy has on the individual and that is why she is betrayed by Sandy. Sandy realizes the destructive nature Miss Brodie has on individual girls and she tries to stop her. Sandy has had an approach/avoidance relationship with Miss Brodie, but in the end, the avoidance takes precedent and Sandy betrays Miss Brodie to save other "individual" girls from being harmed. Miss Brodie eventually loses her job over her support of fascism, so it is an ironic twist in the story. The one thing she admires proves to be her undoing.
Read a good analysis of the novel here on enotes.
Miss Brodie in The Prime of Miss Jean Brodie is a fascist. She holds fascist political beliefs, praises fascist leaders, teaches fascism to her girls, and behaves as a fascist leader toward them. All of the above make her a symbol of fascism.
The ideas of absolute power and absolute authority and absolute efficiency appeal to her. Thus, she is drawn to fascist leaders. That's what she wants among her girls. Everything done in the exact way she wants. That's her sordid idea of what existence should be. She even tries to control her girls' sex lives. She wants to live vicariously through her girls and turn them into little Miss Brodies.