Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C1 |
No aliases for this group.
| Sylow | C3 ⋊ #7441 |
136 conjugacy classes of subgroups; only the 39 normal subgroups are shown.
This group has 136 conjugacy classes of subgroups — too many to draw. Only its 39 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 96 |
Polycyclic presentation:
2^8,3|a^2=d,b^2=e,c^2=fgh,d^2=gh,f^2=h,b^a=bc,c^a=ce,e^a=eg,f^a=fh,i^a=i^2,c^b=cf,d^b=defg,g^b=gh,d^c=dg,e^c=eh51 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 57192);