Finite Group DatabasebetaFinite Group Databasebeta| order | 486 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C32 · Schur multiplier (H2) = C3 |
No aliases for this group.
| Fitting, Sylow | #6910 ⋊ C2 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C3 | C3×(C3⋊D3)⋊C3 |
| commutator, cosocle, hypocenter | H(3,𝔽3) | C2×C32 |
| Frattini | C32 | C32×D3 |
| Fitting | #6910 | C2 |
| prime radical | C3×#272 | C3 |
| hypercenter | C32 | (C3⋊D3)⋊C3 |
| LCS | G ▷ H(3,𝔽3) | |
| UCS | C1 ◁ C3 ◁ C32 | |
76 conjugacy classes of subgroups.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 27 |
A minimal faithful permutation representation — 2 generators on 27 points:
(2,6,8)(3,9)(5,14,16)(7,20,19,17,10,26)(11,23)(12,21)(15,27,18,24,22,25);(1,19,27,13,7,24,4,10,25)(2,22,23,6,15,21,8,18,9)(3,17,5,11,20,16,12,26,14)Polycyclic presentation:
2,3^5|b^3=f,d^a=d^2,e^a=e^2f^2,c^b=cf,d^b=de,e^d=ef^234 conjugacy classes — view the full character table.
| maximal subgroup of | (#4176⋊C2)⋊C3, C2×#92283, #13091112, … |
| maximal normal subgroup of | (#4176⋊C2)⋊C3, C2×#92283, #13091112, … |
| maximal quotient of | #11757587, C2×#92283, #13090756, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(486, 128);