Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Sylow | C3 ⋊ #8507 |
736 conjugacy classes of subgroups; only the 123 normal subgroups are shown.
This group has 736 conjugacy classes of subgroups — too many to draw. Only its 123 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=f,b^2=g,d^2=h,e^2=g,f^2=g,b^a=bd,c^a=ce,d^a=dg,i^a=i^2,d^b=dh,e^b=egh,f^b=fgh,i^b=i^2,d^c=dgh,e^c=eg,f^c=fg78 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 62016);