Yes, the simple fact is, I am afraid, yes. Nazis murdered even children including infants just born with their gas chambers and other methods. That is an awful thing to discuss.
This was the central theme of the Nazis, their entire scheme to completely exterminate Jewish people. So they pursued all the individuals, regardless of their age. It did not matter to them whether one was old, young or a small child.
As the trains with people reached death camps such as Auschwitz, the guards quickly did a sorting. This was what they referred to as the selection. Nearly all children below perhaps 14 years were kidnapped immediately. The Nazis believed themselves to be individuals, who were not able to work and termed them as useless.
These children were directly sent to gas chambers. Their primary, fastest method of killing the large masses of people was the poison gas.
In case of the mothers who gave birth to a child within the camps, the child was almost always killed immediately. The Nazis did not desire any of the Jewish babies to be born and raised. They killed the babies in bad and cruel ways, such as, giving them poison shots or even drowning them.
It was not only gassing. There were also hundreds of thousands of children who were shot or slowly died of no food (starvation) and sickness as the conditions of living in the camp were horrible. It has been estimated that nearly 1.5 million children were killed in this horrible period.