Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #19552 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C24 |
| direct product | C5×#19584 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | C2×#121 |
| commutator, Frattini, cosocle | C42 | C24×C5 |
| prime radical | C5×#3212 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C2×C42×C5 ◁ G | |
584 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)(7,11)(9,16)(12,17)(13,18)(15,20)(19,21)(22,23,27,31)(24,28,33,37)(25,30,26,35)(29,34,36,32);(6,7,9,11,10,13,16,18)(8,12,15,17,14,19,20,21)(22,24)(23,28)(25,34)(26,32)(27,33)(29,35)(30,36)(31,37);(6,8)(7,12)(9,15)(10,14)(11,17)(13,19)(16,20)(18,21)(22,24)(23,29)(25,32)(26,34)(27,33)(28,35)(30,37)(31,36);(7,13)(8,14)(11,18)(15,20)(22,25,27,26)(23,30,31,35)(24,32,33,34)(28,29,37,36)200 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 12572);