Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3057 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C5 · Schur multiplier (H2) = C28 |
| direct product | C5×#60682 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C5 | C22×D4 |
| commutator, Frattini, cosocle | C2×C4 | C25×C5 |
| socle | C22×C5 | C23×D4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C5 ◁ C2×C5×#117 ◁ G | |
1878 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)(16,21)(17,29)(18,23)(19,33)(20,24)(22,31)(25,38)(26,35)(27,36)(28,42)(30,37)(32,43)(34,41)(39,45)(40,44);(7,10)(11,13)(14,16)(15,21)(17,25)(18,27)(19,28)(20,26)(22,34)(23,36)(24,35)(29,38)(30,40)(31,41)(32,39)(33,42)(37,44)(43,45);(14,17,19,31,20,32,18,30)(15,22,23,29,24,37,33,43)(16,25,28,41,26,39,27,40)(21,34,36,38,35,44,42,45);(6,8,9,12)(7,11,10,13)(16,26)(21,35)(25,39)(27,28)(34,44)(36,42)(38,45)(40,41);(14,18,20,19)(15,23,24,33)(16,27,26,28)(17,30,32,31)(21,36,35,42)(22,29,37,43)(25,40,39,41)(34,38,44,45)320 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 53670);