Finite Group DatabasebetaFinite Group Databasebeta| order | 486 |
| isoclinism | isoclinic class of AGL(1,ℤ/27) |
| automorphism | Aut(G) = AGL(1,ℤ/27) (order 486) · Inn(G) = G · Out(G) = C1 |
| homology | Abelianization (H1) = C2×C9 · Schur multiplier (H2) = C1 |
| affine general linear | AGL(1,ℤ/27) |
| unambiguous semidirect product | #6875⋊C2 |
| Fitting, Sylow | #6875 ⋊ C2 |
| derived | C27 ⋊ C2×C9 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C1 | G |
| commutator, hypocenter | C27 | C2×C9 |
| Frattini | C3×C9 | C3×D3 |
| Fitting | #6875 | C2 |
| socle | C3 | D9⋊C9 |
| prime radical | D27⋊C3 | C3 |
| cosocle | C27⋊C3 | C2×C3 |
| LCS | G ▷ C27 | |
| UCS | C1 | |
28 conjugacy classes of subgroups.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 27 |
A minimal faithful permutation representation — 2 generators on 27 points:
(2,22,27,14,23,15,20,8,19,5,18,26,7,21,17,16,6,25)(3,11,12,9,10,24)(4,13);(1,17,14,24,18,21,3,2,25,13,27,16,11,6,8,10,20,26,4,19,5,9,23,22,12,7,15)Polycyclic presentation:
2,3^5|b^3=c,d^3=e^2f^2,e^3=f,d^a=d^2e,e^a=e^2f^2,f^a=f^2,d^b=de,e^b=ef^2,d^c=df31 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(486, 31);