Study of specific Quotients of a $p$-group in MAGMA

Question

In Magma, I wanted to do a program, but since I was getting errors in "basic" commands, I am in trouble. Here I will describe a problem to be done in MAGMA, then its interpretation in MAGMA. I would like to know if the commands are correct.

Problem to be done in MAGMA: $G$ is a $p$-group of order $p^5$ such that $Z(G)=C_p\times C_p\times C_p$ and $\mho_1(G)=\langle g^p\colon g\in G\rangle\cong C_p\times C_p$. For each subgroup $N$ of order $p$ in $Z(G)$ which is not in $\mho_1(G)$, I want to study some properties of the quotient $G/N$.

Commands (program) in MAGMA: I gave the presentation of group (i.e. took $G$ in MAGMA in terms of some generators and relations; it is group of order $p^5=7^5$. This is not troubling me. The trouble is coming in next commands.

  Z:=Center(G);
  A:=Agemo(G,1);
  L:=Subgroups(Z: OrderEqual:=7);
  M:=Subgroups(A: OrderEqual:=7);
  T:=L diff M;

Here, I am doing - list subgroups of $Z(G)$ of order $p=7$ in $L$, then list subgroups of $\mho_1(G)$ of order $p=7$ in $M$, then remove $M$ from $L$, which is set $T$. Now I want to study structure of quotient group $G/N$ for each subgroup in $T$. Here size of $T$ is $(7^2+7+1)-(7+1)=7^2$. Thus, I was doing following:

  #Center(G/T[1]);
  #DerivedSubgroup(G/T[1]);
  #AutomorphismGroup(G/T[1]);

Where is problem? The problem I was facing in the command T:=L diff M. Since I found error, I didn't (and couldn't) run last three commands. I didn't get any solution in this book of MAGMA.

Question: What is the correct way to determine $T$ and to proceed with next commands? Are they correct?

Answer 1

An abelian group is not a good example, because subgroups are the same as conjuagcy classes of subgroups. Here is some code for a group of order $243$ that you might find helpful.

 > G:=SmallGroup(243,50); 
 > A:=Agemo(G,1);
 > L:=[s`subgroup: s in Subgroups(G:OrderEqual:=3)];
 > LL:= &cat[[ l^t : t in RightTransversal(G,Normalizer(G,l))] : l in L];
 > LL:=SequenceToSet(LL);
 > M:=[s`subgroup: s in Subgroups(A:OrderEqual:=3)];
 > MM:= &cat[[ l^t : t in RightTransversal(G,Normalizer(G,l))] : l in M];
 > MM:=SequenceToSet(M);
 > T:=LL diff MM;