No. According to Nietzsche, the übermensch (superman) has to be of what he termed “master morality” and not “slave morality.” Those of master morality do not need any codes or laws to guide them because they have their own internal ones - their own moral framework. This means those of master morality will step on others (those of slave morality) and crush them to actualise themselves, obtain their desires and transcendent truth. They do not flip religious and political mythologies, codes and laws and then use that as a moral framework to live by because in so doing they are still a slave. Only they are wearing a red shirt instead of a blue one.
Those of slave morality want what those of master morality have, and exalt and praise their own weaknesses. They are envious and their entire reason for asserting themselves is due to envy - to being inferior - for wanting to become the master instead of the slave.
“no longer accepting my low place in society. I have made a sworn oath to myself that I will never again accept my role as a lowly class human being.”
This kind of statement and assertion is called “slave revolt.” When you metaphorically overthrow the master you will still be a slave sitting in the master’s chair. Just as an Untermensch will always be an Untermensch regardless of if it assumes the title of “übermensch”. Because what defines the übermensch is not titles or position, but strength, will to power and biological prerequisites (intellect, intelligence, genetics, ferocity etc.). The übermensch is natural superiority fully realised (“superior” in the sense of Nietzsche’s classification). Modern science may be able to alter body chemistry but there are some biological factors that cannot be changed. Not a single one of those characteristics I mentioned above defines “übermensch” on their own. They have to all be present within the individual.
The Superman was regarded by Nietzsche as the evolution of man. He believed man is a tightrope between the animal and the Superman, hanging over the abyss. And that we cannot simply turn back, we must keep going towards the Superman or fall into the abyss trying. Those who are not destined (in accordance with the prerequisites I mentioned) for Superman-hood are to cultivate and assist in any way they can with the ascension of those who are if we are to advance as the human race and the respective sub-races.
In short, the key to collective evolution is elitism with the knowledge that the inferior (not meant in a derogatory term just in the sense that they do not possess the prerequisites to be regarded as superior) are a vital and key component in the ascension of the superior to übermensch, and must be utilised by the superior regardless of if the inferior consent to it or not.
“The strong do what they will, the weak suffer what they must.” - Thucydides
"What is great in man is that he is a bridge and not a goal: what is lovable in man is that he is an OVER-GOING and a DOWN-GOING.
I love those that know not how to live except as down-goers, for they are the over-goers.
I love those who do not first seek a reason beyond the stars for going down and being sacrifices, but sacrifice themselves to the earth, that the earth of the Superman may hereafter arrive.
I love him who labors and invents, that he may build the house for the Superman, and prepare for him earth, animal, and plant: for thus seeks he his own down-going.
I love all who are like heavy drops falling one by one out of the dark cloud that lowers over man: they herald the coming of the lightning, and succumb as heralds.
Lo, I am a herald of the lightning, and a heavy drop out of the cloud: the lightning, however, is the SUPERMAN.”
– Friedrich Nietzsche, Thus Spoke Zarathustra