Finite Group DatabasebetaFinite Group Databasebeta| order | 864 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Sylow | C33 ⋊ #137 |
230 conjugacy classes of subgroups; only the 35 normal subgroups are shown.
This group has 230 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) = 48 |
Polycyclic presentation:
2^5,3^3|b^2=d,d^2=e,b^a=bd,c^a=ce,d^a=de,f^a=f^2,g^a=f^2g,h^a=h^2,c^b=ce,f^b=f^2g^2,g^b=f^2g,f^d=f^2,g^d=g^242 conjugacy classes — view the full character table.
| maximal subgroup of | #13165564, #13165566, #13167528, … |
| maximal normal subgroup of | #13165564, #13165566, #13167528, … |
| maximal quotient of | #13167410, #13167416, #13167426, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(864, 4319);