nls singular gradient error

Question

I want to fit this model of population growth:

$$y(t) = α / (1 + βe^{−γt})$$

given the data:

time<-c (1.000,2.000,3.000,4.000,5.000,6.000,7.000,8.000,9.000,10.000,11.000,12.000)
population<-c (5.308,7.240,9.638,12.866,17.069,23.192,31.443,38.558,50.156,62.948,75.995,91.972)

I attempted to fit it into nls function with starting values as:

alpha=220; beta=50; gamma=0.3

and successfully got:

nls.fit<-nls(population~(alpha)/(1+(beta*exp(-gamma*x.pred))) , data=mydata, start=unscaled.init.values)

with output:

#Nonlinear regression model
#
#  model: population ~ (alpha)/(1 + (beta * exp(-gamma * x.pred)))
#
#   data: mydata
#
#   alpha     beta    gamma 
#
#196.1862  49.0916   0.3136 
#
# residual sum-of-squares: 2.587
#
#Number of iterations to convergence: 3 
#
#Achieved convergence tolerance: 4.493e-06

But then when I tried to rescale the parameter values as:

alpha'=100*alpha; beta'=10*beta; gamma'=0.1*gamma

with the function:

nls.fit2<-nls(population~0.01*alpha/(1+0.1*beta*exp(-10*gamma*x.pred)) , data=mydata, start=scaled.init.values)

and starting values as:

alpha=2.2; beta=5; gamma=3

I received the error:

#Error in nlsModel(formula, mf, start, wts) : 
#singular gradient matrix at initial parameter estimates

I don't understand why this happens as I only rescaled the parameters, but I also rescaled the starting values to fit the model, so the two nls() should give me the same output? Can someone please explain?