Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #465 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C5×C8 · Schur multiplier (H2) = C23 |
| direct product | C5×#11584 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C5×C8 | C2×D4 |
| commutator | C2×C4 | C22×C5×C8 |
| Frattini, cosocle | C4×C8 | C23×C5 |
| socle | C22×C5 | C2×C22⋊C8 |
| prime radical | C5×#532 | C4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C5×C8 ◁ C2×C4×C5×C8 ◁ G | |
402 conjugacy classes of subgroups.
| minimal degree | μ(G) = 45 |
| transitive degree | μt(G) = 640 |
A minimal faithful permutation representation — 3 generators on 45 points:
(1,2,3,4,5)(6,7)(8,10)(9,12)(11,13)(14,15,17,22,18,23,27,35,19,21,28,33,29,34,39,43)(16,20,25,30,26,31,38,40,24,32,36,41,37,42,44,45);(7,10)(8,13)(11,12)(14,16)(15,20)(17,25)(18,26)(19,24)(21,32)(22,30)(23,31)(27,38)(28,36)(29,37)(33,41)(34,42)(35,40)(39,44)(43,45);(15,21)(16,24)(22,33)(23,34)(25,36)(26,37)(35,43)(38,44)440 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 4572);