Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1304 |
| automorphism | ? |
| homology | Abelianization (H1) = C4×C5×C8 · Schur multiplier (H2) = C4 |
| direct product | C5×#7197 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C5 | #95 |
| commutator | C23 | C4×C5×C8 |
| Frattini, cosocle | C2×C4×C8 | C22×C5 |
| socle | C22×C5 | #336 |
| prime radical | C23×C5 | C4×C8 |
| LCS | G ▷ C23 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C5 ◁ C2×C4×C5×C8 ◁ G | |
184 conjugacy classes of subgroups.
| minimal degree | μ(G) = 53 |
| transitive degree | μt(G) = 640 |
A minimal faithful permutation representation — 2 generators on 53 points:
(1,2,3,4,5)(6,7)(8,14,15,11)(9,13)(10,12)(16,20,21,18)(17,19)(22,23,25,29,26,31,36,43,27,30,37,42,38,44,49,52)(24,32,34,45,35,46,48,53,33,28,39,40,47,41,50,51);(6,8,9,16,10,15,17,21)(7,11,13,18,12,14,19,20)(22,24)(23,28,30,32)(25,34)(26,35)(27,33)(29,40,42,45)(31,41,44,46)(36,48)(37,39)(38,47)(43,51,52,53)(49,50)260 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 185);