Finite Group DatabasebetaFinite Group Databasebeta| order | 864 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Sylow | H(3,𝔽3) ⋊ AGL(1,ℤ/8) |
278 conjugacy classes of subgroups; only the 50 normal subgroups are shown.
This group has 278 conjugacy classes of subgroups — too many to draw. Only its 50 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 72 |
Polycyclic presentation:
2^5,3^3|b^2=d,d^2=e,b^a=bd,d^a=de,g^a=g^2,h^a=h^2,c^b=ce,f^b=f^2,h^b=h^2,g^f=gh49 conjugacy classes — view the full character table.
| maximal subgroup of | #13137453, #13137456, #13139018, … |
| maximal normal subgroup of | #13137453, #13137456, #13139018, … |
| maximal quotient of | #13137311, #13137314, #13137318, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(864, 1038);