I'd agree with most of those here in advising you to give the required notice. Having been in the post for such a short time they may only wanta short notice period or maybe just a few days depending on the work. Something worth knowing is that if THEY decide that that you don't need to cover the full notice period then they are obliged to pay you for the full period i.e. 'payment in lue of notice'.
In a similar situation I handed in my notice with an accompanying request for a meeting "
to discuss how we work through the notice period" In the meeting I basically asked the person in question how they felt I could usefully use this time. (first I made sure I had already tied up all loose ends) by the end of the conversation, they could find no logical reason for me to work the notice period, so they offered to pay me a months salary in lue of notice. SWEET! But it must be the companies decision otherwise they have no obligation to pay it - don't suggest it yourself!
On a second point, a referance is extremely valuable, trying to find a postion without being able to put your last employers down as references just means one more difficuly in finding a job, even if you've been to collage in between. I managed to get a job in the early years by using a referance from my saturday job I had while I was at school.
A month is a worthy price to pay for a reference. It won't make or break your career, but every and any advantage is worth having in the job market. (and the self discipline is never to early to practice - still working on mine

)
Good luck whatever you decide
Noir UK