Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C42 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| Sylow | C3 ⋊ #7316 |
620 conjugacy classes of subgroups; only the 67 normal subgroups are shown.
This group has 620 conjugacy classes of subgroups — too many to draw. Only its 67 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,b^2=e,b^a=bc,c^a=cf,e^a=eg,g^a=gh,c^b=cg,d^b=df,f^b=fh,i^b=i^2,e^d=eh60 conjugacy classes — view the full character table.
| maximal subgroup of | #72231556, #72231567, #72231586, … |
| maximal normal subgroup of | #72231556, #72231567, #72231586, … |
| maximal quotient of | C2×#10682486 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 56867);