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) ⋊ #135 |
239 conjugacy classes of subgroups; only the 42 normal subgroups are shown.
This group has 239 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) = 144 |
Polycyclic presentation:
2^5,3^3|b^2=e,c^2=de,d^2=e,c^a=cd,d^a=de,g^a=g^2,h^a=h^2,c^b=cd,d^b=de,f^b=f^2,h^b=h^2,g^f=gh52 conjugacy classes — view the full character table.
| maximal subgroup of | #13137003, #13137005, #13137006, … |
| maximal normal subgroup of | #13137003, #13137005, #13137006, … |
| maximal quotient of | #13138749, #13138761, #13138763, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(864, 811);