I was in exactly the same position as you with £550 from Scottish power nine months after leaving, and I managed to argue them around and didn't pay a penny. You have to show three things:
One) You didn't realise you were overpaid. I did this by saying I gave my legally required notice, they said they'd adjust my final pay, they did, its hardly my fault they didn't do it right.
Two) It wasn't my fault, it was theirs. Largely the same as above, I'd given notice, their pay dept are the ones to blame here and crucially, I had no reason to suspect I was overpaid.
Three) It'd cause you financial hardship to pay the money back- I said I was a full time student with no income, and given the above, insisting I hand over the cash was unfair.
Now, in these circumstances, if you make an offer to pay back the money at say £10 a month, they cannot refuse it. They also cannot call in debt collectors having refused a repayment offer- it wouldn't stand up in court. The guy that sits next to me at the football for the last 10 years is an employment lawyer, so he wrote them a letter saying I'd taken legal advice, stated the above, that I knew where I stood and they couldn't proceed with trying to collect the debt. They wrote me a reply saying they had dropped any attempts to pursue it. Happy days.