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All finite 2-groups G with the following property are classified: Property. The Frattini subgroup of G contains an abelian subgroup of rank 3, but G contains no normal abelian subgroup of rank 3. The method of classification involves showing that if G is such a group, then G contains a normal abelian subgroup W isomorphic to Z4 x Z4, and that the centralizer C of W in G has an uncomplicated structure. The groups with the above property are then constructed as extensions of C. Introduction. In recent years there has been interest in the class of finite 2-groups which contain no normal abelian subgroup of rank three (cf. [3]). The 2-groups which contain no normal abelian subgroup of rank two have been known for some time. In classifying those 2-groups which have no normal abelian subgroup of rank two, one may first show that if G is such a group, then the Frattini subgroup of G contains no elementary abelian subgroup of order 4 [2, Satz III, 7.5]. Next one shows that G has a cyclic subgroup of index 2 [2, Satz III, 7.6], and then one may easily construct the 2-groups which have no normal elementary abelian subgroup of order 4 [2, Satz I, 14.9b]. Therefore, as a first step in the classification of 2-groups G which contain no normal abelian subgroup of rank three, it seems reasonable to consider groups G with the following property: (*) G is a 2-group and ^(G), the Frattini subgroup of G, contains an elementary abelian subgroup of order 8, but G contains no normal elementary abelian subgroup of order 8. In this paper we classify all groups with property (*). Such a group will be called a (*)-group. The paper is divided into four sections. In the first section we discuss the general structure of 2-groups satisfying (*). We show that if G is such a group, then <b(G) contains a subgroup W which is normal in G, and W is isomorphic to the direct Presented to the Society, January 22, 1971 under the title A class of 2-groups; received by the editors May 7, 1971. AMS 1969 subject classifications. Primary 2040, 2027.

$2$-groups of normal rank $2$ for which the Frattini subgroup has rank $3$