Finite Group DatabasebetaFinite Group Databasebeta| order | 864 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Sylow | C27 ⋊ #137 |
120 conjugacy classes of subgroups; only the 47 normal subgroups are shown.
This group has 120 conjugacy classes of subgroups — too many to draw. Only its 47 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 432 |
Polycyclic presentation:
2^5,3^3|a^2=e,d^2=e,f^3=g,g^3=h,c^a=cd,d^a=de,f^a=f^2g^2h^2,g^a=g^2h^2,h^a=h^2,c^b=ce,d^c=de141 conjugacy classes — view the full character table.
| maximal subgroup of | #13134330, #13134331, #13134646, … |
| maximal normal subgroup of | #13134330, #13134331, #13134646, … |
| maximal quotient of | #13134502, #13134505, #13134506, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(864, 165);