Is there a rule of thumb about the minimum population required for a viable recovery of a species?

Minimum viable population is a concept people talk about. Falling near or below the MVP is referred to as a population bottleneck and can affect the composition of the gene pool of a species. Species differ so much that it is hard to draw conclusions; cheetahs, for instance, have so little genetic variability that their MVP is likely very low, possibly no greater than 1 breeding pair. Other species are thought to have much higher MVP requirements.
The kakapo (Māori: kākāpō or night parrot), Strigops habroptilus (Gray, 1845), also called owl parrot, is a species of large, flightless, nocturnal, ground-dwelling parrot of the super-family Strigopoidea endemic to New Zealand.

The passenger pigeon is the other end of the spectrum, an example of a species that was not thought to be threatened because it was always found in immense flocks; it seems, though, that passenger pigeons do not do well outside their gigantic flocks - collectively, as a species they may have required some advantage that being in a huge flock gave them. In any event, once a tipping point was reached the species disappeared entirely within a very short period of time, and they are extinct so we will never know for sure.

The California condor bottlenecked down to 27 individuals in captivity and 0 in the wild at one point; there are now estimated to be 435 living specimens, thanks largely to the tireless efforts of conservationists. As birds go, the very interesting Kakapo is another example. It is a flightless parrot - really peculiar bird, quite large, that long relied on insularity. It only inhabits a few rodent-free islands and doesn't do well at all in captivity; I fear we are going to lose the Kakapo forever, in not very long, despite the best efforts of conservationists.

Source
https://en.wikipedia.org/wiki/Minimum_viable_population
https://en.wikipedia.org/wiki/California_condor
https://en.wikipedia.org/wiki/Kakapo

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