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×#32793 |
| 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 | |
592 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)(30,31,34,36)(32,35,40,43)(33,37,41,45)(38,42,39,44);(6,8)(7,10)(9,12)(11,13)(14,16,19,17)(15,20,22,23)(18,25,26,24)(21,28,29,27)(30,32)(31,35)(33,38)(34,40)(36,43)(37,42)(39,41)(44,45);(7,11)(10,13)(14,17,19,16)(15,20,22,23)(18,24,26,25)(21,28,29,27)(32,38)(33,41)(35,42)(37,45)(39,40)(43,44);(8,12)(10,13)(14,18,19,26)(15,21,22,29)(16,24,17,25)(20,27,23,28)(31,36)(35,43)(37,45)(42,44)170 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 25781);