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Let p be an odd prime number. We describe the Whitehead group of allextra-special and almost extra-special p-groups. For this we compute, for anyfinite p-group P , the subgroup Cl\_1 (ZP) of SK\_1 (ZP), in terms of a geneticbasis of P. We also introduce a deflation map Cl\_1 (ZP) $\rightarrow$ Cl\_1(Z(P/N)) , for a normal subgroup N of P , and show that it is alwayssurjective. Along the way, we give a new proof of the result describing thestructure of SK\_1 (ZP), when P is an elementary abelian p-group.

The Whitehead group of (almost) extra-special p-groups with p odd