 Full-vertex modules for a noncyclic p-group

Every noncyclic finite finite p-group $P$ has a quotient $C_p\times C_p$. Over any field of characteristic $p$, define a square-zero representation with basis $v_0,\dots ,v_r,w_1,\dots ,w_r$ by $x{v}_i=w_i$ for $1\le i\le r$, $x{v}_0=0$, $y{v}_i=w_{i+1}$ for $0\le i<r$, $y{v}_r=0$, and $xW=yW=0$. Here $x=g-1$, $y=h-1$, and $W$ is the span of the $w_i$. An endomorphism induces the same scalar on the top and on $W$; the remaining maps from the top to $W$ form a square-zero ideal. Thus the endomorphism ring is local and the module is indecomposable. Inflating to $P$ and choosing $p\nmid 2r+1$ gives infinitely many distinct dimensions with vertex $P$: the trace of a relative-trace identity from any proper subgroup would force dimension zero in the field. This construction works over finite fields too.