8777 is the serial number for the 1908 model. The Steyr factory started a new series with each model, ie 1903, 1905, 1908, 1910, 1924, 1925, each beginning with "1".
25452.19 is the ledger number of the Vienna, NPv, proofhouse. It means your rifle is the gun number 25452 proofed there in 1919, any make or model. Such ledger numbers were only stamped by the proofhouses of the former Austro-Hungarian empire, that is Austrian, Czech and Hungarian.
C8.0 is the caliber, of course.
The number on the outside of the buttstock is unusual. It is neither a factory- nor proofhouse one. It may be a registration number by a former owner?