Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #29602 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C23×C4 |
| direct product | C5×#29724 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C5 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24×C5 |
| prime radical | C24×C5 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C5 ◁ C2×C5×#117 ◁ G | |
922 conjugacy classes of subgroups.
| minimal degree | μ(G) = 49 |
| transitive degree | μt(G) = 640 |
A minimal faithful permutation representation — 4 generators on 49 points:
(1,2,3,4,5)(6,7,8,9)(10,11)(12,15)(13,14)(16,17)(18,19)(20,24)(21,25)(22,27)(23,26)(28,37)(29,39)(30,38)(31,40)(32,42)(33,41)(34,44)(35,45)(36,43)(46,48)(47,49);(11,14)(15,17)(18,20,23,30)(19,24,26,38)(21,31,33,47)(22,34,36,29)(25,40,41,49)(27,44,43,39)(28,35,46,32)(37,45,48,42);(7,9)(10,12)(11,15)(13,16)(14,17)(18,21,23,33)(19,25,26,41)(20,28,30,46)(22,35,36,32)(24,37,38,48)(27,45,43,42)(29,47,34,31)(39,49,44,40);(12,16)(15,17)(19,26)(24,38)(25,41)(27,43)(37,48)(39,44)(40,49)(42,45)200 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 22712);