I think in the beggining it will be kinda akward no matter what. At least for you, but its okay to feel nervous, the important thing is to look lowkey confident.

You said you made eye contact with her sometimes, I think this is the beginning. Start interacting quickly with her, for example, when you cross with her in a corridor say hello to her with a hand wave or something, this way you will slowly “break the ice” between you. Also, sometimes you will greet her and she wont notice and you will feel embarassed but its fine, it happens, just dont let it get to you and try to be a nice person. Just live your life and when opportunities like these happen, go for it. This way, when you finally decide to approach her to talk it wont be coming out of nowhere.

Be nice to her friends too, even if you think you wont fit, its a good idea to also start interacting with them, you may even find out you guys do have some things in common and it will be easier to discover what kind of thing she likes etc. I like to keep in mind that relationships (romantic and friendship ones) are built. Their feeling towards you and your relationships are not something that exist by itself, they have their roots in material reality, therefore the idea of all this I’m saying is to slowly increase the amount of “positive interactions” between you and her group, so when you approach her, she will be more receptive.

With this part covered, now she (and her friends) have an idea of who you are and that you are friendly, so talking to her will not be out of nowhere. As for approaching her, is she from your classroom? If so, things are easier, you can ask her about somethibg school related just as an introduction and try to keep the conversation from there. Every time you talk things will get easier and easier. But if she is not from your classroom then you should seek something else. Try something regarding your surroundings that relate to both of you. For example, if you both take the same bus to school you could start the conversation asking about the if the bus has passed (it hasnt obviously, she is right there) and start talking about it, what she thinks of it, and from there there are an infinite amount of topics to go through, it will depend on what she says and how engaged in the conversation she is. (Please remember not to push it if she is refusing to talk or is disintereted and let it go, it happens). If she talks about taking the bus to go to the swimming class, you can shift the conversation to swimming class, because it is something she brought up and it will be way more interesting than bus.

Here are some tips on the talking itself: -If she engages in the conversation, the best way to keep it going and avoid that akward “dead end” is to make questions she need to “develop” to awnser. Like:

Do you like XiJinping?

Yes is the anwser. But that’s it. The conversation dies. Unless you follow up with:

What did you like the most about his speech at the event of 100 years of the CPC?

Her awnser will not be a simple word and will probably end with “I can’t wait for the fall of the american empire”.

I know your focus is to get involved romanticaly with her but this is useful for talking to anyone you want to be friends with. This is great because it means you can “practice” befriending and approaching other people before talking to her. Honestly talking to people in general has always been hard for me but these ideas helped a lot. It doesnt matter your gender or sexual orientation either.

EDIT: I read someone saying romanticism is inherently liberal and I agree, so Im calling the thing I described here the “Dialetical romantic materialism”. We shall win the hearts of everyone on earth with our material analysis of love comrades.