Sample size and social mobility for rare surnames

I am an amateur researcher looking at social mobility for surnames held by low status individuals in 19th century Australia. I only have high school statistics so I would like to know a bit about sample sizes.

I have constructed a spreadsheet listing 160 rare surnames held by a total of 512 19th century low status individuals out of a total pool of some 74,000 such low status individuals.

I then looked up the rare surnames in the 2017 online landline phone book and found 983 entries.

I then looked up all the rare surnames in online directories of lawyers and general practice doctors (two “high status” occupations) and found 3 entries.

I then looked at a set of 19 common surnames that were probably neither high or low status in the 19th century. These names produced 9278 phone book entries and 73 lawyers/doctors.

My question is, how large a sample do I need to build to give me some confidence that the difference in rates of people moving into the two high status occupations actually means something and is not just random chance?