The sound γ loses its voicing before ς = [ks] = ξ, as in fox, box, etc. ξ is a consonant cluster that can begin a word, so it falls under item b below. See
http://perseus.uchicago.edu/cgi-bin/philologic/getobject.pl?c.9:1.NewPerseusMonographs section §140.
Here are Smyth's rules:
In pronouncing Greek words and in writing (at the end of the line) the rules commonly observed are these:
a. A single consonant standing between two vowels in one word belongs with the second vowel: ἄ-γω, σο-φί-ζω.
b. Any group of consonants that can begin a word, and a group formed by a stop with μ or ν, and by μν, belongs with the second vowel: τύ-πτω, ὄ-γδοος, ἄ-στρον, ἔ-χθος; πρᾶ-γμα, ἔ-θνος, λί-μνη.
c. A group of consonants that cannot begin a word is divided between two syllables: ἄν-θος, ἐλ-πίς, ἔρ-γμα. Doubled consonants are divided: θάλατ-τα.
d. Compounds divide at the point of union: εἰς-φέρω, προς-φέρω; ἀν-άγω, εἰσάγω, συν-έχω. (But the ancients often wrote ἀ-νάγω, εἰ-σάγω, προ-σελθεῖν, ἐ-ξάγω, δυ-σάρεστος.)
e. ς, when followed by one or more consonants, is either attached to the preceding vowel (ἄ-ρις-τος), or, with the consonant, begins the following syllable (ἄ-ρι-στος). (The ancients were not consistent, and there is evidence for the pronunciation ἄ-ρις-στος.)
f. The ancients divided ἐκ τούτου as ἐ-κ τού-του. This practice is now abandoned.