Finite Group DatabasebetaFinite Group Databasebeta| order | 486 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6856 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C32 · Schur multiplier (H2) = C3 |
| direct product | C2×#6861 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C32 | H(3,𝔽3) |
| commutator, Frattini, cosocle | C33 | C2×C32 |
| prime radical | C2×C33 | C32 |
| LCS | G ▷ C33 ▷ C32 ▷ C1 | |
| UCS | C1 ◁ C2×C32 ◁ C2×C33 ◁ G | |
50 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 162 |
A minimal faithful permutation representation — 2 generators on 56 points:
(1,2)(4,7,13)(5,10,20)(8,17,27)(9,16,24)(11,22,21)(14,18,23)(15,19,26)(25,28,29)(30,31,34,33,37,42,40,47,53)(32,36,41,38,46,52,48,44,54)(35,43,49,45,51,50,55,56,39);(3,4,8,6,9,18,12,19,28)(5,7,14,10,16,25,20,26,27)(11,15,17,21,13,23,22,24,29)(31,35,44)(32,38,48)(34,41,51)(36,37,45)(39,50,49)(42,52,56)(43,53,54)(46,47,55)70 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(486, 68);