Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24378 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C2×C42 |
| direct product | C5×#29553 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23×C5 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24×C5 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C5 ◁ C43×C5 ◁ G | |
732 conjugacy classes of subgroups.
| minimal degree | μ(G) = 37 |
| transitive degree | μt(G) = 1280 |
A minimal faithful permutation representation — 4 generators on 37 points:
(1,2,3,4,5)(6,7)(8,12)(9,11)(10,13)(22,23,25,28,26,29,34,37)(24,30,31,35,32,36,33,27);(6,8,9,13)(7,10,11,12)(14,15,17,18)(16,19,20,21);(15,18)(19,21)(22,24,26,32)(23,27,29,35)(25,33,34,31)(28,36,37,30);(7,11)(10,12)(14,16,17,20)(15,19,18,21)290 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 22541);