Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C42 · Schur multiplier (H2) = C4 |
No aliases for this group.
| Sylow | C3 ⋊ #7222 |
214 conjugacy classes of subgroups; only the 87 normal subgroups are shown.
This group has 214 conjugacy classes of subgroups — too many to draw. Only its 87 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 192 |
Polycyclic presentation:
2^8,3|a^2=d,b^2=e,c^2=fgh,e^2=fh,f^2=h,b^a=bc,c^a=cf,e^a=efg,f^a=fh,i^a=i^2,c^b=ch,d^b=dg,i^b=i^284 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 56621);