cfdenger:
The number 56237 is the serial number. It is always stamped on the bottom of the grip frame and sometimes in other locations.
The number 17857 is probably an assembly number, numbers stamped on fitted together parts as they worked their way through the assembly process.
In 1955 the small 'J frame' S&Ws in .38 S&W special caliber used the serial range 55050 to 75000. Because of the serial number it sounds like you have a .38 special caliber Chiefs Special, but the barrel should be marked .38 S&W SPL.
If your revolver has a 'hump backed' look with the hammer enclosed except for the very tip of the hammer spur, so that when it is cocked only the spur is outside the frame, then it is probably a Bodyguard or Airweight Bodyguard, but these have a higher serial number range than your gun, serial number 66000 being recorded in the Standard Catalog of S&W as the starting point.
The last possibility, and most likely, is that you have a post war .38 Regulation Police. These are five shot revolvers chambered for the .38 S&W caliber, not .38 special, using the slightly smaller 'I frame'. These, like the Chiefs Special, have an exposed hammer, when it is cocked you can see the firing pin from the side. The Regulation Police was marked '.38 S&W CTG' on the right side of the barrel and had a small S&W trademark on the left side of the frame.
Postwar production started in 1948 at serial number 54474.
In 1957 the Chiefs Special was renamed the model 36, the Bodyguard was renamed the model 49, the Airweight Bodyguard was renamed the model 38 & the Regulation Police was renamed the model 33.
MY guess is you have a post war Regulation Police and based on the serial number it was probably made in the first couple of years of production, from 1948 to 1950 seems likely.
I eagerly await your photos to see if I am right!
