Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Fitting | C3×#1813 ⋊ C2 |
| Sylow | C3 ⋊ #13076 |
466 conjugacy classes of subgroups; only the 123 normal subgroups are shown.
This group has 466 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) = 96 |
Polycyclic presentation:
2^8,3|a^2=f,b^2=dfg,d^2=gh,f^2=h,g^2=h,b^a=bd,c^a=ce,d^a=dg,e^a=eh,g^a=gh,e^b=eh,i^b=i^2,d^c=dh,f^c=fh78 conjugacy classes — view the full character table.
| maximal subgroup of | #72270389, #72270418, #72270422, … |
| maximal normal subgroup of | #72270389, #72270418, #72270422, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 87025);