No, the (ancient) Greek is γένεσις.
For some reason unknown to me, some words from this root have one ν (γένεσις, γενιά = generation (as in grandparents - parents - children), γενέθλια = birthday), while other words have double ν (γέννηση, γεννώ = give birth, γεννήτρια = generator)
Whether the -ν is used in the end of some words or not, such as neuter nouns or adjectives ending in -ο(ν) depends on the version of the language that is used. In older versions, the correct form used to be δώρον (gift) or χόρτον (grass, weed). In modern Greek, though, the correct is δώρο and χόρτο. I guess you might have seen Χόρτον written in an old sign?
Something similar happens with the masculine and feminine accusative in singular, where the correct form of the article used to be τον and την only, but in modern Greek the -ν is omitted whenever the next letter isn't a vowel or plosive consonant. This rule is clear, i.e. you can always decide the correct form, whereas with the derivatives of γεννώ this is not the case.