Finite Group DatabasebetaFinite Group Databasebeta| order | 864 |
| isoclinism | isoclinic class of D27 |
| automorphism | ? |
| homology | Abelianization (H1) = C32 · Schur multiplier (H2) = C1 |
| unambiguous semidirect product | C27⋊C32 |
| derived, Sylow | C27 ⋊ C32 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C16 | D27 |
| commutator, hypocenter | C27 | C32 |
| Frattini | C9×C16 | D3 |
| Fitting, cosocle | C16×C27 | C2 |
| socle, prime radical | C2×C3 | C9⋊C16 |
| LCS | G ▷ C27 | |
| UCS | C1 ◁ C16 | |
24 conjugacy classes of subgroups.
| minimal degree | μ(G) = 59 |
| transitive degree | μt(G) = 864 |
Polycyclic presentation:
2^5,3^3|a^2=b,b^2=c,c^2=d,d^2=e,f^3=g,g^3=h,f^a=f^2g^2h^2,g^a=g^2h^2,h^a=h^2240 conjugacy classes — view the full character table.
| maximal subgroup of | C32×D27, #13133943, C2×C27⋊C32, … |
| maximal normal subgroup of | C32×D27, #13133943, C2×C27⋊C32, … |
| maximal quotient of | C27⋊C64, C2×C27⋊C32 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(864, 1);