Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
344 conjugacy classes of subgroups; only the 35 normal subgroups are shown.
This group has 344 conjugacy classes of subgroups — too many to draw. Only its 35 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 64 |
Polycyclic presentation:
2^5,3,2^3|a^2=d,c^2=i,g^2=i,h^2=i,b^a=bc,c^a=cei,f^a=f^2,g^a=gh,h^a=hi,c^b=ci,d^b=de,g^f=h,h^f=ghi,h^g=hi44 conjugacy classes — view the full character table.
| maximal subgroup of | #72268131, #72268142, #72268160, … |
| maximal normal subgroup of | #72268142, #72268160, #72271322, … |
| maximal quotient of | #72268855, #72268859, #72268906, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1085439);