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) |
292 conjugacy classes of subgroups; only the 42 normal subgroups are shown.
This group has 292 conjugacy classes of subgroups — too many to draw. Only its 42 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=e,d^2=e,c^a=cd,d^a=de,g^a=g^2,h^a=h^2,c^b=cde,d^b=de,f^b=f^2,h^b=h^2,d^c=de,g^f=gh40 conjugacy classes — view the full character table.
| maximal subgroup of | #13149915, #13149919, #13149921, … |
| maximal normal subgroup of | #13149915, #13149919, #13149921, … |
| maximal quotient of | #13137863, #13137879, #13137880, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(864, 1160);