Which conjugation is Gnōthi ‘know’, as in Gnōthi sauton ‘know thyself’?

This is the aorist imperative active, 2nd person singular, of γιγνώσκω ‘to know’

Alas, γιγνώσκω ‘to know’ is one of the many irregular verbs of Greek. The particular irregularity here is that while its present tense is thematic (a normal -ω verb), it forms its aorist stem γνω- according to the older, athematic paradigm (represented by verbs whose present ends in -μι). So this is an archaic aorist imperative ending, where “normal” verbs have -ε instead.

Smyth’s Grammar, Herbert Weir Smyth, A Greek Grammar for Colleges, goes into the history of these forms—and you need to, for cases like this.

466. ENDINGS OF THE IMPERATIVE

1. Active.

a. 2 Sing.—λῦε, λίπε, τίθει (for τίθε-ε) have not lost –θι. –θι is found in 2 aor. pass. φάνη-θι; in στῆ-θι and ἕστα-θι; in some 2 aorists, like γνῶ-θι, τλῆ-θι, πῖ-θι, which are μι forms though they have presents of the ω form (687). Also in ἴσ-θι be or know, ἴθι go, φάθι or φαθί say. λύθητι is for λυθηθι by 125 b.

466 a. D. –θι is not rare in Hom., pres. δίδωθι ῀δίδου, ὄρνυθι, aor. κλῦθι, perf. τέτλαθι. Aeolic has ἴστα_, φίλη. πίει, δέχοι, δίδοι (Pindar) are very rare.

Let’s take this slowly. The normal ending of the imperative 2nd sg is -ε. The older ending is -θι, and you still see it in places in Homer, where Classical Greek would use -ε instead. The old -θι is preserved in the 2nd aorist passive [EDIT: and the 1st aorist passive, where -θη-θι gets dissimilated to -θη-τι]; it is also preserved in the aorist imperative for “stand, know, go, say”, which are athematic verbs (present ἵστημι, [οἶδα], εἶμι, φημί). And it is also preserved in a few 2nd aorists which use old athematic forms “know, suffer, drink”.

Yes, these are irregularities. Sorry. Like Desmond James says, the useful thing to do here is not so much to memorise every verb, as to get familiar with the range of possible endings: just know that -θι is an archaic imperative ending, and you can work out the details later. To identify γνῶθι as an aorist, you rule out the present tense stem, because you know that is reduplicated: γί-γνω-σκε. So γνω- is, by default, the aorist instead.

Yes. I know. Sorry.