Finite Group DatabasebetaFinite Group Databasebeta| order | 486 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of H(3,𝔽3) |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C34 · Schur multiplier (H2) = C35 |
| direct product | C2×C3×#737 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C3×C9 | C32 |
| commutator, Frattini, cosocle | C3 | C2×C34 |
| socle | C2×C32 | C33 |
| prime radical | C2×C3×H(3,𝔽3) | C3 |
| LCS | G ▷ C3 ▷ C1 | |
| UCS | C1 ◁ C2×C3×C9 ◁ G | |
480 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 162 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2)(3,4,5);(7,10,16)(11,19,17)(12,18,24)(20,27,25)(21,26,30)(28,32,31);(6,7,11)(8,12,20)(9,10,17)(13,21,28)(14,18,25)(15,16,19)(22,26,31)(23,24,27)(29,30,32);(6,8,13,9,14,22,15,23,29)(7,12,21,10,18,26,16,24,30)(11,20,28,17,25,31,19,27,32)198 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(486, 253);