Giant radio halos are the most relevant examples of diffuse synchrotron emission from galaxy clusters .
A number of these sources have very steep spectrum , with spectral index \alpha \geq 1.5 - 1.6 ( F ( \nu ) \propto \nu ^ { - \alpha } ) , and are ideal targets to test current models for the origin of the relativistic particles .
A2256 hosts the nearest radio halo with very steep spectrum , with \alpha = 1.61 , and a very large population of relativistic protons in the cluster would be necessary to explain the halo as due to synchrotron emission from secondary particles .
In this case the 0.1-1 GeV \gamma –ray luminosity is expected 10-20 times larger than that of clusters hosting radio halos with similar radio power at GHz frequencies but with spectra more typical of the presently observed halo population , \alpha \sim 1.2 .
Under these assumptions incoming FERMI/GLAST observations are expected to detect A2256 , provided that the magnetic field in the central cluster region is \leq 10-15 \mu G. We show that this will allow for a prompt test of hadronic models for the origin of radio halos , and for complementary constraints on both the cluster magnetic field and the physics of particle acceleration mechanisms .