1. I tried to find a reference for the computation of the Schur multiplier of groups of order $p^4$.
The case in which $p=2$ is well known, see e.g. Table 1 at <http://pages.bangor.ac.uk/~mas010/pdffiles/non-abel-tensor.pdf>.
However, I didn't find any reference for the odd case.
Since there is a well known (not too big) classification of such groups, it seems reasonable that this work was already done.
Is there a paper or a book in which I can find such computation?
2. I am particularly interested in special elements of the Schur multiplier.
These elements of $M(G)$ correspond to projective representation of $G$ of dimension $\sqrt{|G|}$. Groups admitting such elements are called of central type (non-classical).
Is there a known classification of groups of central type of order $p^4$?


Thanks in advance.



Two questions on the Schur multiplier of groups of order $p^4$
If I understand the question, the problem is to find all groups $X$ of order $p^4$ such
that $X \cong P/Z(P)$ for some group $P$, where $P$ has an irreducible character of degree $p^2$. To do this, it seems that we need consider only groups $P$ of order $p^5$ or $p^6$. I have done this by "brute force" for $p = 3$, using the Magma software. Of the 15 groups of order $3^4$, exactly five have the desired property. They are numbers 2,4,9,12 and 15 of the small groups data base. 


A group $X$ is often called a "central-type factor group" (CTFG) if
$X \cong P/Z(P)$, where $P$ has an irreducible character of degree $\sqrt |X|$. There are various things known about CTFGs, and in particular, an abelian group is a CTFG if and only if it has the form $A \times B$, where $A \cong B$. It is also known that a CTFG cannot have a self-centralizing cyclic subgroup. (This condition eliminates five of the 15 groups of order $3^4$.)


