Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33517 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C24 |
| direct product | C5×#33676 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | C22×D8 |
| commutator, Frattini, cosocle | C2×C8 | C24×C5 |
| prime radical | C2×C5×#135 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C2×C42×C5 ◁ C2×C4×C5×C8 ◁ G | |
548 conjugacy classes of subgroups.
| minimal degree | μ(G) = 45 |
| transitive degree | μt(G) = 640 |
A minimal faithful permutation representation — 4 generators on 45 points:
(1,2,3,4,5)(6,7)(8,11)(9,10)(12,13)(14,15,20,25)(16,21,29,37)(17,23,19,33)(18,22,32,39)(24,30,38,31)(26,35,28,43)(27,34,42,45)(36,40,44,41);(7,10)(11,13)(14,16,20,29)(15,21,25,37)(17,26,19,28)(18,27,32,42)(22,34,39,45)(23,35,33,43)(24,36,38,44)(30,40,31,41);(15,22)(17,19)(18,31)(21,34)(23,24)(25,39)(26,28)(27,41)(30,32)(33,38)(35,36)(37,45)(40,42)(43,44);(6,8,9,12)(7,11,10,13)(14,17,20,19)(15,23,25,33)(16,26,29,28)(18,30,32,31)(21,35,37,43)(22,38,39,24)(27,40,42,41)(34,44,45,36)230 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 26664);