Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #60454 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C5 · Schur multiplier (H2) = C27×C4 |
| direct product | C5×#60620 |
| 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 | |
2014 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)(16,26,28,37)(17,31,32,39)(18,24,19,23)(21,30,36,29)(22,33,38,34)(27,35,41,45)(40,42,44,43);(7,10)(11,13)(14,16,19,30,20,28,18,29)(15,21,24,37,25,36,23,26)(17,27,34,43,32,41,33,42)(22,35,31,44,38,45,39,40);(6,8,9,12)(7,11,10,13)(14,17,18,33,20,32,19,34)(15,22,23,39,25,38,24,31)(16,27,29,42,28,41,30,43)(21,35,26,40,36,45,37,44);(17,32)(22,38)(27,41)(31,39)(33,34)(35,45)(40,44)(42,43);(16,28)(21,36)(26,37)(27,41)(29,30)(35,45)(40,44)(42,43)290 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 53608);