I have 3 groups I want to study using a single independent variable. In particular, I wish to know the sample size needed for each group. In the case of having 2 groups, it seems power analysis while estimating effect size is the correct approach; we compute effect size as Cohen's $d = (\mu_1-\mu_2)/s_p^2$ where $\mu_i$ is the mean of group $i$ and $s_p^2$ is the pooled variance. We then provide a value for $\alpha$ and power (often denoted as $1-\beta$) and from here we can conclude the sample size needed.
But I don't know how to perform power analysis for 3 (or more groups). What's more it seems that Tukey's pair wise HSD test allows me to accept or reject, pair-wise, whether a given pair is distinct or not once I have effect size, sample size, etc.
I would like to know how big each group needs to be and I suppose after I run the experiment I can then run Tukey's test. Lastly, I did run a small experiment with 8 samples for group 1, 8 for group 2, and 20 for group 3; so I believe these could be used for estimating the effect size to help determine what the size for the sample should be in the next (larger) study.