Abstract: In the group testing problem, which goes back to the work of Dorfman (1943), we aim to identify a small set of $k\sim n^\theta$ infected individuals out of a population size $n$, $0<\theta<1$. We avail ourselves to a test procedure that can test a group of individuals, with the test returning a positive result iff at least one individual in the group is infected. All tests are conducted in parallel. The aim is to devise a test design with as few tests as possible so that the infected individuals can be identified with high probability. We establish an explicit sharp information-theoretic/algorithmic phase transition $\minf$, showing that with more than $\minf$ tests the infected individuals can be identified in polynomial time, while this is impossible with fewer tests. In addition, we obtain an optimal two-stage adaptive group testing scheme. These results resolve problems prominently posed in [Aldridge et al. 2019, Johnson et al. 2018, M\'ezard and Toninelli 2011]