There is a functor F: {finite nilpotent Algebra over a finite field F} > {finite pgroups} sending J to 1+J.(You can think of J as the Jacobsonradical of 1F+J) and sending f:A>B to F(f):1+A > 1+B with F(f)(1+a)=1+f(b). I want to know which classes of pgroups are realisable as 1+J and maybe if you see some interesting properties of the functor F. For example if J^2 = 0,then G=1+J is elementary abelian and if J^3=0 then G=1+J is special. Such realisations could be interesting since,we have for example in general G' $\leq $ 1+J^2 and a series 1+J $\leq$ 1+J^2 ... $\leq$ 1+J^k =0 and there it is known that the characters of 1+J are induced from linear characters of a subgroup 1+A where A is a subalgebra of J.

3$\begingroup$ These groups are sometimes called algebra groups  this might help your search. There's a great deal of literature on their character theory. $\endgroup$ – M T Jun 18 '11 at 13:42

$\begingroup$ thank you.do you know a paper which contains most of the known results about those groups?by the way: if someone has an interesting example of a pgroup not realisable as 1+J,I would be glad if you post it. $\endgroup$ – trew Jun 18 '11 at 15:54
For algebras over nonprime fields, a good reference is Isaacs (1995). It proves a fundamental theorem establishing that these Falgebra groups behave like Fgroups, not just like pgroups, for char(F) = p. I think these groups have been studied over Z/pZ or even Z/p^{n}Z for a very long time.
As far as a pgroup not of the form 1 + J: any cyclic pgroup of large enough exponent will do. For instance, the cyclic group of order 8 is not of the form 1 + J: P = J, and so J has dimension 3 over F = Z/2Z. Hence 1F+1J has dimension 4, and P is contained in a Sylow 2subgroup of GL(4,2), and hence has exponent dividing 4. Similar bounds hold for odd p: the cyclic group of order p^{2} is not an algebra group, for instance. Every group of order dividing 8 (other than the cyclic group of order 8) is an algebra group.
I assume you have already seen the use of prestricted Lie algebras in finite pgroups. This takes a pgroup and associates a Lie algebra with pretty similar properties as J has in 1+J. If not, then this may be exactly what you are looking for and is very well studied (refs on requests, but any "good book on pgroups" should work).
 Isaacs, I. M. Characters of groups associated with finite algebras. J. Algebra 177 (1995), no. 3, 708–730. MR1358482 DOI:10.1006/jabr.1995.1325

1$\begingroup$ The groups of order 16 that are not algebra groups are exactly the cyclic, the maximal class (D16, QD16, Q16), and the modular group. The groups of order p^3 (p odd) that are not algebra groups are the cyclic and the extraspecial group of exponent p^2. I am nervous about p^4: it seems (too) rare to be an algebra group. $\endgroup$ – Jack Schmidt Jun 19 '11 at 14:22