In this note, we give the result of a computer search for the smallest fullerene that does not allow a face spiral code as used by Manolopoulos and Fowler and adopted in IUPAC recommendations for fullerene nomenclature. The search enumerated all the small fullerenes on up to 400 vertices and the conclusion is that the smallest fullerene without a face spiral has 380 vertices. (C) 2011 Elsevier B. V. All rights reserved.