Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Sylow | C3 ⋊ #7515 |
210 conjugacy classes of subgroups; only the 27 normal subgroups are shown.
This group has 210 conjugacy classes of subgroups — too many to draw. Only its 27 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 48 |
Polycyclic presentation:
2^8,3|a^2=d,e^2=gh,f^2=h,b^a=bc,c^a=ce,e^a=ef,f^a=fg,g^a=gh,i^a=i^2,d^b=degh,e^b=egh,f^b=fgh,d^c=dfg,e^c=egh,f^c=fh,e^d=egh,f^d=fh48 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 57384);