Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3068 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C5 · Schur multiplier (H2) = C25 |
| direct product | C5×#21559 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C5 | C2×#121 |
| commutator | C2×C4 | C23×C4×C5 |
| Frattini, cosocle | C22×C4 | C24×C5 |
| socle | C2×C5 | C4×#121 |
| prime radical | C5×#3067 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C5 ◁ C22×C5×C8 ◁ G | |
868 conjugacy classes of subgroups.
| minimal degree | μ(G) = 37 |
| transitive degree | μt(G) = 160 |
A minimal faithful permutation representation — 4 generators on 37 points:
(1,2,3,4,5)(6,7,11,17,13,18,12,16)(8,19,10,14,22,29,21,30)(9,23,25,15,26,31,27,32)(20,34,24,33,35,28,36,37);(6,8,11,10,13,22,12,21)(7,14,16,19,18,30,17,29)(9,20,25,24,26,35,27,36)(15,28,23,33,32,34,31,37);(6,9)(7,15)(8,20)(10,24)(11,25)(12,27)(13,26)(14,28)(16,23)(17,31)(18,32)(19,33)(21,36)(22,35)(29,37)(30,34);(6,10,11,22,13,21,12,8)(7,14,17,29,18,30,16,19)(9,24,25,35,26,36,27,20)(15,28,31,37,32,34,23,33)290 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 14547);