My first guess was that it was putrescine and other polyamines---but the smell of putrefying potatoes actually comes from methyl mercaptan, dimethyl sulfide, and dimethyl trisulfide. Check it:
Source: A. Kamiya, Y. Ose, Study on offensive odor (Report IV): A Consideration of Putrefaction and Offensive Odor of Solid Waste, Journal of Japan Society of Air Pollution, 18(5), 1983, pp 453-463.
Notice that the headspace composition changes quite a bit with time. Most of the headspace chromatography studies I found dealt with early detection of disease organisms in infected potatoes, rather than potatoes in full putrefaction mode. You'll also find many references to solanine poisoning from potatoes; solanine is a toxic glycoalkaloid, is nonvolatile, and has nothing at all to do with the foul smell and toxic gas produced by putrid potatoes.
Methyl mercaptan ($\rm CH_3SH$) has an odor described as "rotting cabbage" by ATDSR; it's one of the major contributors to the smell of farts (oh, sorry, "flatus" if we're being polite). It has an odor detection threshold as low as 1 ppb.
Dimethyl sulfide ($\rm (CH_3)_2S$) is responsible for the smell of the sea (in low concentrations); it too has a cabbagy smell.
Dimethyl trisulfide ($\rm CH_3SSSCH_3$) is present in relatively smaller amounts but it has an even stronger odor. The detection threshold is around 1 part per trillion, and it is apparently a strong insect attractant.
The gases produced by rotting potatoes are quite toxic (see Rotting potatoes in basement kill four members of Russian family).
(To keep the notes below in context, in my original answer I said that a plot point in "The Walking Dead" was that zombies couldn't smell delicious humans if they were wearing coats smeared with rotting flesh. I suggested that when the zombie apocalypse arrives, packing your pockets with putrid potatoes might work, too. Now I'm not so sure. Can zombies distinguish between polyamines and sulfur compounds? Perhaps they're stench connoisseurs.)