Profusely flowering cactus

I was given this magnificent cactus plant, a Mammillaria species, but I don't know which one. It has several pendulous 'branches', and each has a circle of flowers at the end. One 'branch' has flowers part way down the column as well. The flowers have lasted for ages, opening every day and closing at night.

It is great to have the conservatory back, now our house repairs are completed (well, the inside, anyway), and I'm looking forward to acquiring a few new stone plants to replace the ones that overheated while we were out of the house. We brought them back to the conservatory while the builders were still here because we found ants in our temporary accommodation, and didn't want to transfer any home with us. Unfortunately the builders were very security conscious, and closed up the conservatory completely so it got very warm, too hot for even these desert plants!