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×#19585 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | C2×#121 |
| commutator, Frattini, cosocle | C42 | C24×C5 |
| prime radical | C5×#121 | D4 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C2×C42×C5 ◁ G | |
550 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)(6,7,11,15)(8,16,18,19)(9,13,12,21)(10,17,20,14)(22,23,27,31)(24,28,33,37)(25,30,26,35)(29,34,36,32);(6,8,10,13,11,18,20,21)(7,12,14,16,15,9,17,19)(22,24)(23,28)(25,34)(26,32)(27,33)(29,35)(30,36)(31,37);(6,9)(7,13)(8,17)(10,19)(11,12)(14,18)(15,21)(16,20)(22,24)(23,29)(25,32)(26,34)(27,33)(28,35)(30,37)(31,36);(8,18)(9,12)(13,21)(16,19)(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, 12573);