Hello Singledad,
Welcome. I too am heartened to hear how much you care about your daughter and how you want to help her. It sounds like you already are in a very powerful way that many of us underestimate: you are listening. You are actively listening. This is so important. Just having permission to say what she needs to say will help your daughter feel stronger and more real.
There are ways to communicate your support for your daughter without undermining her relationship with her mother. And that is by allowing your daughter to voice how she is feeling about how things are going at home. How does she really feel about what's going on? What are her choices next time x,y,z happens?
I have also learned the hard way from my daughter that sometimes she just wants me to listen and does NOT want me to get involved because that can make things more complicated. (I'm not divorced, I'm speaking of her issues with classmates...) You can encourage her to have friends over (they make wonderful witnesses and/or shields to hold off N behavior, since Ns like to look good for the outside world). She can ask to be taken to the library for a school project. Different strategies to get away from places where abuse happens.
You can give her tools to use such as phrases to say to anyone who mistreats or disrespects her (this way you are not targeting her mother).
Something else I tell my kids is that all families have issues and their classmates have lessons they are struggling with too. I don't say this to be competitive or to deflect. I say it so they know they are not inferior or superior, just on the same playing field with everyone else. Well, she is only six, so I might hold off on that discussion, but my point is all kids need love in order to face the challenges life is going to throw them.
I hope this helps a little. Your little girl is very fortunate to have you looking out for her. Good luck, MP