Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32712 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C23 |
| direct product | C5×#32795 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C5 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24×C5 |
| prime radical | C24×C5 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C5 ◁ C2×C5×#117 ◁ G | |
584 conjugacy classes of subgroups.
| minimal degree | μ(G) = 45 |
| transitive degree | μt(G) = 640 |
A minimal faithful permutation representation — 4 generators on 45 points:
(1,2,3,4,5)(6,7)(8,10)(9,11)(12,13)(14,15,19,22)(16,23,17,20)(18,21,26,29)(24,28,25,27)(30,31,34,38)(32,39,41,36)(33,42,35,37)(40,45,43,44);(6,8)(7,10)(9,12)(11,13)(14,16,19,17)(15,20,22,23)(18,25,26,24)(21,28,29,27)(32,40)(33,35)(36,44)(37,42)(39,45)(41,43);(7,11)(10,13)(14,17,19,16)(15,20,22,23)(18,24,26,25)(21,28,29,27)(30,32)(31,36)(33,40)(34,41)(35,43)(37,45)(38,39)(42,44);(8,12)(10,13)(14,18,19,26)(15,21,22,29)(16,24,17,25)(20,27,23,28)(30,33,34,35)(31,37,38,42)(32,40,41,43)(36,45,39,44)170 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 25783);