# Multinomial logistic regression using glmnet

I have a few questions regarding the use of the glmnet package. I have a data with n observation, p variables and k classes. I use the command

cv.glmnet(x, y, family="multinomial")


to fit my data in r. What exactly does $nzero tell us? is it the number of none-zero coef period? ie. variable p2 in class k3 is a none-zero coef, so it counts, or is it the number of variable that has a none-zero coef in any class. I'm asking this since when I do the grouped version: cv.glmnet(X, Y, family="multinomial", type.multinomial = "grouped")  it seems to be the latter, ie. the number of variables. I know that in grouped lasso variable are all in or not at all. However the number of coefs I have left in the none-grouped model are suspiciously low and I want to be sure. I've also noticed that certain lambda value produce even better %dev (using cvfitg$glmnet.fit) yet it is not chosen to be lambda.min. So if the %dev is not the CVed min then what is it?