Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #25140 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C24 |
| direct product | C5×#25265 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | C2×#121 |
| commutator, Frattini, cosocle | C22×C4 | C24×C5 |
| prime radical | C5×#525 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C23×C4×C5 ◁ G | |
728 conjugacy classes of subgroups.
| minimal degree | μ(G) = 45 |
| transitive degree | μt(G) = 320 |
A minimal faithful permutation representation — 4 generators on 45 points:
(1,2,3,4,5)(6,7)(8,11)(9,10)(12,13)(14,15,19,24,20,25,18,23)(16,21,28,37,29,38,27,36)(17,30,32,40,33,41,34,22)(26,42,43,39,31,45,44,35);(7,10)(11,13)(14,16,19,28,20,29,18,27)(15,21,24,37,25,38,23,36)(17,31,32,44,33,26,34,43)(22,39,30,45,40,35,41,42);(6,8,9,12)(7,11,10,13)(14,17)(15,22)(16,26)(18,32)(19,34)(20,33)(21,35)(23,30)(24,41)(25,40)(27,43)(28,44)(29,31)(36,42)(37,45)(38,39);(7,10)(11,13)(14,18,20,19)(15,23,25,24)(16,27,29,28)(17,32,33,34)(21,36,38,37)(22,30,40,41)(26,43,31,44)(35,42,39,45)215 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 18253);