I want to determine how many non-isomorphic extensions (as group they are non-isomorphic) are possible of the form $1 \to \mathbb{S}^1 \to G \to (\mathbb{Z}\_p)^k \to 1$, where $G$ is a compact lie group and $\Bbb{Z}\_p$ is the cyclic group of order $p$? 
The only possible homomorphism $(\Bbb{Z}\_p)^k \to Aut(\mathbb{S}^1)=\Bbb{Z}\_2$ is the trivial one if $p\neq 2$. If $p=2$, then we have $(\Bbb{Z}\_2)^k$ possible homomorphism $(\Bbb{Z}\_2)^k \to \Bbb{Z}\_2$. Then we have to calculate the group $H^2((\Bbb{Z}\_p)^k; \mathbb{S}^1)$. I don't know how to proceed further. A detailed answer will be very helpful.


Thank you very much in advance.



How many non-isomorphic extensions with kernel $S^1$ and quotient cyclic of order $p$?
Let's only consider central extensions (so we only miss a few cases when $p=2$). I denote by $C\_p=\mathbf{Z}/p\mathbf{Z}$, to avoid confusion with $p$-adics.


So extensions are classified by $H^2(C\_p^k,S^1)$. The commutator map yields a canonical homomorphism $\phi$ from $H^2(C\_p^k,S^1)$ onto $\mathrm{Hom}(\Lambda^2C\_p^k,S^1)$, and the latter is isomorphic to $C\_p^{k(k-1)/2}$ (more canonically, the dual of $\Lambda^2C\_p^k$). The kernel of this homomorphism consists of those 2-cocycle defining an abelian extension; since $S^1$ is an injective $\mathbf{Z}$-module the only abelian extension is split so $\phi$ is an isomorphism. Hence, the extensions are classified by $C\_p^{k(k-1)/2}$, more precisely the space of alternating bilinear forms on $C\_p^k$.


Next, the isomorphism class of the groups thus obtained are classified by the quotient of the latter by the action of $\mathrm{GL}\_k(\mathbf{Z}/p\mathbf{Z})$. This quotient has exactly $1+\lfloor k/2\rfloor$ elements (number of types of alternating forms in dimension $k$, regardless of the field, including characteristic 2). 


Note: For $k=1,2,3$ this makes $1$, $2$, $2$ isomorphism types of Lie groups. That for $k=2,3$ it does not match with Ian's answer (3, 8) is because he also considers quotients of order $p^k$ that are not $p$-elementary.


