Geodesic
The Modular Group Algebras of P-Groups of Maximal Class
Canadian journal of mathematics, Tome 40 (1988) no. 6, pp. 1422-1435

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The isomorphism problem for modular group algebras of finite p-groups appears to be still far from a solution (see [7] for a survey of the existing results). It is therefore of interest to investigate the problem for special classes of groups.The groups we consider here are the p-groups of maximal class, which were extensively studied by Blackburn [1]. In this paper we solve the modular isomorphism problem for such groups of order not larger than p p+1, having an abelian maximal subgroup, for odd primes p.What we in fact do is to generalize methods used by Passman [5] to solve the isomorphism problem for groups of order p 4. In Passman's paper the case of groups of maximal class is actually the most difficult one. 
Bagiński, C.; Caranti, A. The Modular Group Algebras of P-Groups of Maximal Class. Canadian journal of mathematics, Tome 40 (1988) no. 6, pp. 1422-1435. doi: 10.4153/CJM-1988-065-2
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