Finite \(p\)-groups up to isoclinism, which have only two conjugacy lengths (Q1806106)

Let \(G\) be a finite group and let \(\text{Cl}(G)\) be the set of conjugacy classes of \(G\). Let \(\text{cl}(G)=\{|C|:C\in\text{Cl}(G)\}=\{1,n_1,\dots,n_r\}\), where \(|C|\) is the number of elements in \(C\) and \(1<n_1<\dots<n_r\). Then \(G\) is called a group of conjugacy type \(\{1,n_1,\dots,n_r\}\). The structure of \(p\)-groups of conjugacy type \(\{1,n\}\) is studied here. The classification of \(p\)-groups of conjugacy type \(\{1,p\}\) and \(\{1,p^2\}\) is given using Hall's concept of isoclinism which is an equivalence relation between groups and it preserves the conjugacy type.