# [C (gcc)], <s>790 778 757 748 731</s> 687 bytes

<!-- language-all: lang-c -->

    typedef struct F F;typedef struct{int c,p;F**m;}g;struct F{int d;g f;};g S,y,q,*x=&y,*p=&q;A(g*g,F*s){g->c++[g->m=g->c-g->p?g->m:realloc(g->m,(g->p+=32)*20)]=s;}i,*a,c[999],b,j,k,l;*B(d){F*s=0;for(i=S.c;i--;)s=S.m[i]->d-d?s:S.m[i];!s?s=calloc(5,4),s->d=d,A(&S,s):0;return s;}F*u,*v;E(){for(i=-1;b=++i<S.c;b&&A(x,S.m[i]))for(l=0;k=l<x->c&b;b&=!k)for(j=0,u=x->m[l++];j<u->f.c&&(k=u->f.m[j++]->d-S.m[i]->d););x->c>p->c?a=x,x=p,p=a:0;x->c=0;}main(){for(;~scanf("%d %d",&a,&b)&&A(&(u=B(a))->f,v=B(b))|~A(&v->f,u););for(E();i=b<S.c;)c[b]<b?a=S.m[i=b%2*c[b]++],S.m[i]=S.m[b],E(S.m[b]=a):(c[b++]=0);for(printf("largest clique size: %d\nclique members:",p->c);i<p->c;)printf(" %d",p->m[i++]->d);}

This is a slightly golfed version of the reference implementation. 
[Try it online!](https://tio.run/##VVJdb@IwEHznV6RIRHayqQhfBxiDWqn8AR45HhInRIGEhpggehz963TXpj3dQ9bjmcnubBQVZErd76ePKk3SraNPdaNOztJZiv@pa344OQoqsfS8Utwy8e00QiIyZytuWFfwAUfwLtL9AK@S7lG8sMzLYOlpfs2CufL9NR6lJBxgqRZ0ndZpVBTvitEFqFa@7Pe41@vyjdTiloMXgVpPJpMNxLCDPRTCe2UJv2Jn2RXb95rlcvWsRB4EgmuE5TrfBPMkSBZ6am/iSS@0VHbSEAYcNBpkAi/MXYHm066o01NTHxycuPQa8M7ijfGrbR6EIpa@n89oSuy6L@wCti/n5Cgwxl4Wswtu5sbokE97I@xkFxqJdLkufH8jdrMmmG@fleuyvTSwXO9QoLA/sbnggjrNKyyLSF7gIiuoZIQhicdhtzLKD4944lOr6LBl7U7idJI2uBG4MaeQLmvkK4s4x0FwRhhz/vcT@TMRDc2hBrinyGVsluNqHW9mMU41cWTc6XlEYcbHxkaIN/DGLJARnzK0oEN2bcOqxj8DAxVRnaUaf54iPzapo/M/6RQj/j48iDIt47TW0zbQphhiRqfg3@@bdSr6eLn9Rlzc7vewNxk5/XDUCkfDrgXE0M2iYc9Kv8Kwhc8/LwrDnqGNjwDeScYjHI6GLSoOiRb9vGlFM6Y/fri@genXH9sANM8oRAwGLXyccDx6JLMIC0Ux1lF/8gU "C (gcc) – Try It Online")

[C (gcc)]: https://gcc.gnu.org/

Slightly golfed less

    typedef struct F F;
    // student group
    typedef struct{
      int c, // count
          p; // capacity
      F**m;  // members
    }g;
    // student
    struct F{
      int d; // id
      g f;   // friends
    };
    g S,y,q,*x=&y,*p=&q;
    A(g*g,F*s){
      g->c++[g->m=g->c-g->p?g->m:realloc(g->m,(g->p+=32)*20)]=s;
    }
    i,*a,c[999],b,j,k,l;
    *B(d){
      F*s=0;
      for(i=S.c;i--;)
        s=S.m[i]->d-d?
          s
        :
          S.m[i];
      !s?
        s=calloc(5,4),
        s->d=d,A(&S,s)
      :
        0;
      return s;
    }
    F*u,*v;
    E(){
      for(i=-1;b=++i<S.c;b&&A(x,S.m[i]))
        for(l=0;k=l<x->c&b;b&=!k)
          for(j=0,u=x->m[l++];j<u->f.c&&(k=u->f.m[j++]->d-S.m[i]->d););
      x->c>p->c?a=x,x=p,p=a:0;
      x->c=0;
    }
    main(){
      for(;~scanf("%d %d",&a,&b)&&A(&(u=B(a))->f,v=B(b))|~A(&v->f,u););
      for(E();i=b<S.c;)
        c[b]<b?
          a=S.m[i=b%2*c[b]++],
          S.m[i]=S.m[b],
          E(S.m[b]=a)
        :
          (c[b++]=0);
      for(printf("largest clique size: %d\nclique members:",p->c);i<p->c;)
        printf(" %d",p->m[i++]->d);
    }