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×#25271 |
| 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)(14,15,20,33,19,24,18,23)(16,21,34,45,27,37,26,36)(17,28,30,22,32,39,31,40)(25,41,42,35,29,44,43,38);(7,10)(11,13)(14,16,18,26,19,27,20,34)(15,21,23,36,24,37,33,45)(17,29,31,42,32,25,30,43)(22,38,28,44,40,35,39,41);(6,8,9,12)(7,11,10,13)(14,17,19,32)(15,22,24,40)(16,25,27,29)(18,30,20,31)(21,35,37,38)(23,39,33,28)(26,42,34,43)(36,44,45,41);(7,10)(11,13)(14,18,19,20)(15,23,24,33)(16,26,27,34)(17,30,32,31)(21,36,37,45)(22,39,40,28)(25,42,29,43)(35,44,38,41)215 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 18259);