The answer to this depends on a couple of things. There is a 50/500 https://www.britannica.com/science/50-500-rule rule that we can more or less take as a estimation. The people who live on this island came from somewhere else so their starting population could have been larger than 50 but probably not 500, although that is speculation. So if they had 50 people and they had traditions and taboos about marrying children or siblings they should be able to survive fairly well without horrible mutations. Although they will have mutations, but those bad mutations will not survive like say a Habsburg did without a large infrastructure and people tending to them so it will not enter the population permanently.
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