Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1504 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C5×C8 · Schur multiplier (H2) = C23 |
| direct product | C5×#11470 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C5 | C2×C22⋊C4 |
| commutator | C2×C4 | C22×C5×C8 |
| Frattini, cosocle | C22×C8 | C23×C5 |
| socle | C22×C5 | C2×C22⋊C8 |
| prime radical | C5×#525 | C4 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C5 ◁ C23×C5×C8 ◁ G | |
392 conjugacy classes of subgroups.
| minimal degree | μ(G) = 45 |
| transitive degree | μt(G) = 320 |
A minimal faithful permutation representation — 3 generators on 45 points:
(1,2,3,4,5)(6,7)(8,12,13,10)(9,11)(14,15,19,25,20,26,33,40,21,27,17,23,18,24,34,41)(16,28,31,44,30,39,45,37,32,42,29,43,35,22,36,38);(6,8)(7,10)(9,13)(11,12)(14,16)(15,22)(17,31)(18,35)(19,29)(20,30)(21,32)(23,38)(24,42)(25,37)(26,28)(27,39)(33,36)(34,45)(40,43)(41,44);(14,17,20,34,21,19,18,33)(15,23,26,41,27,25,24,40)(16,29,30,36,32,31,35,45)(22,37,28,43,39,38,42,44)320 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 4458);