Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #60367 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C5 · Schur multiplier (H2) = C28 |
| direct product | C5×#60569 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | C23×D4 |
| commutator, Frattini, cosocle | C2×C4 | C25×C5 |
| prime radical | C22×C5×#41 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C2×C5×#117 ◁ G | |
2010 conjugacy classes of subgroups.
| minimal degree | μ(G) = 45 |
| transitive degree | μt(G) = 320 |
A minimal faithful permutation representation — 5 generators on 45 points:
(1,2,3,4,5)(6,7)(8,11)(9,10)(12,13)(14,15,20,25)(16,21,27,35)(17,30,32,39)(18,24,19,23)(22,33,38,31)(26,40,41,45)(28,37,29,36)(34,43,44,42);(7,10)(11,13)(14,16,20,27)(15,21,25,35)(17,26,32,41)(18,28,19,29)(22,34,38,44)(23,36,24,37)(30,40,39,45)(31,42,33,43);(14,17)(15,22)(16,26)(18,31)(19,33)(20,32)(21,34)(23,30)(24,39)(25,38)(27,41)(28,42)(29,43)(35,44)(36,40)(37,45);(6,8,9,12)(7,11,10,13)(16,27)(21,35)(26,41)(28,29)(34,44)(36,37)(40,45)(42,43);(14,18,20,19)(15,23,25,24)(16,28,27,29)(17,31,32,33)(21,36,35,37)(22,30,38,39)(26,42,41,43)(34,40,44,45)290 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 53557);