Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9519 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C5 · Schur multiplier (H2) = C23 |
| direct product | C5×#9527 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | #386 |
| commutator | C22×C4 | C22×C4×C5 |
| Frattini, cosocle | C2×C42 | C23×C5 |
| prime radical | C5×#532 | C22 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C2×C42×C5 ◁ G | |
436 conjugacy classes of subgroups.
| minimal degree | μ(G) = 37 |
| transitive degree | μt(G) = 640 |
A minimal faithful permutation representation — 3 generators on 37 points:
(1,2,3,4,5)(7,11)(8,14)(9,17)(13,19)(15,16)(18,20)(22,23,26,29,27,31,25,30)(24,32,34,36,35,37,33,28);(6,7,9,11,10,13,17,19)(8,12,15,18,16,21,14,20)(22,24)(23,28,31,36)(25,33)(26,34)(27,35)(29,32,30,37);(6,8)(7,12)(9,15)(10,16)(11,18)(13,21)(14,17)(19,20)(23,29)(25,26)(28,32)(30,31)(33,34)(36,37)185 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 2515);