Howdy
I can make out from your photo that the grips cover the bottom of the grip frame.
That makes it a Smith and Wesson 32 Regulation Police, like this one:
Unlike most Smith and Wesson revolvers, the serial number is on the front face of the grip frame, because the bottom of the grip frame is hidden under the wooden grips.
The bottom of the grips should have a patent marking like this:
The metal grip frame on these is actually relatively short. This is how the grips were inletted to fit over the metal grip frame. This grip frame style was patented, hence the patent marking on the bottom of the grips.
This is how the grips look from the rear.
This revolver was chambered for both the 32 S&W Long cartridge and the 38 S&W cartridge. It is built on the S&W I frame, and the cylinder of an I frame revolver is not large enough for six 38 caliber chambers, so the 38s were five shooters. In this photo, a 38 Regulation Police is at the top, a 32 Regulation Police is at the bottom. Notice with the 32 the middle flute on the cylinder is lined up in the middle of the cylinder window in the frame. Notice how the middle flute is wider and off center in the 38, because it is a five shooter instead of a six shooter. Notice too, the 32 is marked 32 LONG CTG, which means 32 S&W Long Cartridge.
The 32 S&W Long cartridge is basically the same as the Colt 32 New Police cartridge. The Colt round has a flat nose and the bullet is 2 grains heavier. They have pretty much the same ballistics.
The 32 Regulation Police was made from 1917 until 1942. Serial numbers ran from about 260,000 to about 536,000.
I have two of them, 3712XX shipped in 1925 and 4083XX shipped in 1924. Not unusual since S&W often shipped out of serial number order.
Your 435556 probably shipped somewhere in that time frame.