Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9431 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C5 · Schur multiplier (H2) = C23 |
| direct product | C5×#9432 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | #386 |
| commutator | C42 | C22×C4×C5 |
| Frattini, cosocle | C2×C42 | C23×C5 |
| prime radical | C5×#3180 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C2×C42×C5 ◁ G | |
436 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)(7,11)(8,15)(9,12)(14,19)(16,17)(18,20)(22,23)(24,31,32,35)(25,28,33,34)(26,29)(27,30)(36,37);(6,7,12,19,10,14,9,11)(8,13,15,20,17,21,16,18)(22,24,26,34,27,32,36,28)(23,25,29,35,30,33,37,31);(6,8)(7,13)(9,16)(10,17)(11,18)(12,15)(14,21)(19,20)(22,25)(23,28)(24,29)(26,35)(27,33)(30,34)(31,36)(32,37)200 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 2420);