Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #17405 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C5 · Schur multiplier (H2) = C24×C4 |
| direct product | C5×#21883 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | C22×C22⋊C4 |
| commutator | C23 | C23×C4×C5 |
| Frattini, cosocle | C22×C4 | C24×C5 |
| prime radical | C24×C5 | C22×C4 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C2×C5×#117 ◁ G | |
954 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)(7,10)(8,12)(14,15,19,23,18,22,32,39)(16,24,27,41,26,20,34,36)(17,28,31,40,30,21,33,38)(25,35,42,44,29,37,43,45);(6,7)(8,11)(9,10)(12,13)(14,16)(15,20)(17,29)(18,26)(19,27)(21,37)(22,24)(23,36)(25,30)(28,35)(31,43)(32,34)(33,42)(38,45)(39,41)(40,44);(7,10)(11,13)(14,17,18,30)(15,21,22,28)(16,25,26,29)(19,33,32,31)(20,35,24,37)(23,40,39,38)(27,43,34,42)(36,45,41,44);(6,8,9,12)(7,11,10,13)(15,22)(16,26)(21,28)(23,39)(25,29)(27,34)(38,40)(42,43)230 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 14871);