ON CERTAIN 2-EXTENSIONS OF Q UNRAMIFIED AT 2 AND ∞

Based on the method of Boston and Leedham-Green et al. for computing the Galois groups of tamely ramified p-extensions of number fields, this paper gives a large family of triples of odd prime numbers such that the maximal totally real 2 extension of the rationale unramified outside the three prime numbers has the Galois group of order 512 and derived length 3. This family is characterized arithmetically, and the explicit presentation of the Galois group by generators and relations is also determined completely.