Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Sylow | C3 ⋊ #12339 |
308 conjugacy classes of subgroups; only the 67 normal subgroups are shown.
This group has 308 conjugacy classes of subgroups — too many to draw. Only its 67 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 384 |
Polycyclic presentation:
2^8,3|a^2=h,c^2=gh,d^2=fh,e^2=gh,f^2=h,b^a=bd,c^a=ce,d^a=df,e^a=egh,f^a=fh,i^a=i^2,d^b=df,e^b=eg,f^b=fh,d^c=dg,e^c=egh60 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 83153);