Dan S
·Great family piece, OP.
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It won’t be a 2577 as those have a screw on back but there was a bumper model with a press on back whose number I forget. 2767 maybe. Or it’s a later 50x model like the 2846. My money is on 2767
Presumably a ref 2765 from 1955 (serial 14.7M). It's a bumper automatic, the specific caliber number is probably hidden under the oscillating weight at the moment. You would need to move it. I don't recall the caliber of the chronometer grade center second bumper movement, maybe 354?