Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| Sylow | C3 ⋊ #12688 |
1220 conjugacy classes of subgroups; only the 123 normal subgroups are shown.
This group has 1220 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=h,b^a=bd,c^a=ce,d^a=dg,g^a=gh,i^a=i^2,f^b=fg,i^b=i^2,f^d=fh72 conjugacy classes — view the full character table.
| maximal subgroup of | #72271675, #72271741, #72271744 |
| maximal normal subgroup of | #72271675, #72271741, #72271744 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 85033);