Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24390 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C23 |
| direct product | C5×#29529 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C5 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24×C5 |
| prime radical | C22×C5×D4 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C5 ◁ C5×#376 ◁ G | |
626 conjugacy classes of subgroups.
| minimal degree | μ(G) = 37 |
| transitive degree | μt(G) = 640 |
A minimal faithful permutation representation — 4 generators on 37 points:
(1,2,3,4,5)(6,7,9,11)(8,12,13,10)(24,30)(25,33)(27,34)(29,35)(31,36)(32,37);(6,8,9,13)(7,10,11,12)(14,15)(16,19)(17,18)(20,21)(22,23,26,28)(24,31,32,27)(25,29,33,35)(30,36,37,34);(15,18)(19,21)(22,24,25,30,26,32,33,37)(23,27,29,34,28,31,35,36);(7,11)(10,12)(14,16,17,20)(15,19,18,21)(23,28)(27,31)(29,35)(34,36)230 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 22517);