Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #60457 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C5 · Schur multiplier (H2) = C27×C4 |
| direct product | C5×#60624 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | C23×D4 |
| commutator, Frattini, cosocle | C2×C4 | C25×C5 |
| prime radical | C5×#3500 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C4×C5×#41 ◁ G | |
2038 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)(16,26)(17,21)(18,23)(19,34)(20,25)(22,29)(24,27)(28,40)(30,35)(31,36)(32,38)(33,44)(37,42)(39,41)(43,45);(7,10)(11,13)(14,16,27,31,20,29,19,17)(15,21,34,22,25,36,24,26)(18,28,41,43,32,42,33,30)(23,35,44,37,38,45,39,40);(6,8,9,12)(7,11,10,13)(14,17,19,29,20,31,27,16)(15,22,24,21,25,26,34,36)(18,30,33,42,32,43,41,28)(23,37,39,35,38,40,44,45);(14,18)(15,23)(16,28)(17,30)(19,33)(20,32)(21,35)(22,37)(24,39)(25,38)(26,40)(27,41)(29,42)(31,43)(34,44)(36,45);(18,32)(23,38)(28,42)(30,43)(33,41)(35,45)(37,40)(39,44)320 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 53612);