Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #10012 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C5 · Schur multiplier (H2) = C23 |
| direct product | C5×#10016 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C5 | #127 |
| commutator, Frattini, cosocle | C2×C42 | C23×C5 |
| socle | C22×C5 | #392 |
| prime radical | C22×C5×D4 | D4 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C5 ◁ C2×C42×C5 ◁ G | |
560 conjugacy classes of subgroups.
| minimal degree | μ(G) = 37 |
| transitive degree | μt(G) = 320 |
A minimal faithful permutation representation — 3 generators on 37 points:
(1,2,3,4,5)(6,7,10,13)(8,14,16,21)(9,17,19,12)(11,18,20,15)(22,23,27,31)(24,32,35,36)(25,30,26,29)(28,34,37,33);(6,8,10,16)(7,11,13,20)(9,18,19,15)(12,21,17,14)(22,24,27,35)(23,28,31,37)(25,34,26,33)(29,36,30,32);(11,20)(14,21)(23,29)(25,26)(28,36)(30,31)(32,37)(33,34)200 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 3004);