Made by Century Records, Pty Ltd, Sydney. Note that the example shown indicayes on the label - Recorded in the U.S.A. by The Carrano Company.

Numbers: There appears to have been three ranges: 1001-104 in 1950; CY1015-27 also in 1950, and a 2000 series of various prefixeas also from 1950. 

Records in collection:

L076 sc3 CENTURY RECORDS  Made by Century Records, Pty Ltd, Sydney.
CY1017 (MX10075).  MOLLY MALONEY'S WEDDING  (O'Brien Boys).  
The Knaves, with Rhythm Accompaniment.
CY1017 (MX10289).  JOLLY OLD ST.NICHOLAS (Vaughn Horton).  
The Knaves, with Nancy Harrie at the Piano.

L115 c3+ CENTURY RECORDS  Made by Century Records Pty Ltd., Sydney.
Recorded in the U.S.A. by The Carrano Company.
CY1010 (MX50-171).  THE TEARS OF ST. ANNE  (Fetters, Porter)
Russ Vincent. With the Lloyd Shaffer Orchestra, Ray Charles Choir.
CY1019 (MX50-170).  SAINT FRANCES CARRINI (Orlob)  Ibid.

Note:
Millr [J] lists two jazz records on the Century label, catalog DD1006, and DD1006/1007all by Jack Allan's Katzenjammers. The leader is on piano, with Ron Tame vocal, Keith Silver, alto and clarinet, Don Andrews guitar, Reg Robinson, string bass, Al Vincer, vibraphone, Franck Marcy on drums - recorded Sydney 23 Fenruary 1950; and one record recorded under Jack Allen and His Band, catalog 1007A/1008A, recorded Sydney April 1950. Note that the leader's name was spelt bith Allen and Allan. These are the only jazz records listed, but of course there may well have been many popular and light instrumental recordings also.