So far when using GAP to find examples/counter-examples, my work flow tends to be to load the small group library for a particular order, and then 'filter' through it to see if what I am looking for exists. Using the SglPPow package I can do this up until the groups of order $3^{8}$. However now I would like to search beyond this, and I am wondering what the 'best' approaches are.
For instance, say now I want to see if a group of order $3^9$ exists with some property. What are my best options? My initial reading suggests ConstructAllGroups in the GrpConst package (and maybe PqDescendants in the ANUPQ package may be useful to me.
To make the question more concrete, in particular I am interested in the class 2 groups of order $3^{9}$ with center of size $3^{4}$ and $Z(G) \cong C_{p^{2}}\times C_{p} \times C_{p} $. How would you approach constructing these?
Thank you