Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33752 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C23×C4 |
| direct product | C5×#33778 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | C22×D8 |
| commutator, Frattini, cosocle | C2×C8 | C24×C5 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C2×C42×C5 ◁ C2×C4×C5×C8 ◁ G | |
598 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)(8,12)(9,11)(10,13)(14,15)(16,26)(17,25)(18,21)(19,32)(20,23)(22,29)(24,34)(27,42)(28,44)(30,37)(31,39)(33,41)(35,40)(36,43)(38,45);(6,8,9,13)(7,10,11,12)(15,21)(16,27)(23,25)(24,37)(28,35)(29,43)(32,39)(33,45);(8,13)(10,12)(14,16)(15,22)(17,28)(18,27)(19,33)(20,35)(21,36)(23,38)(24,40)(25,41)(26,32)(29,34)(30,43)(31,45)(37,44)(39,42);(7,11)(10,12)(14,17,18,20)(15,23,21,25)(16,28,27,35)(19,30,31,34)(22,38,36,41)(24,39,37,32)(26,40,42,44)(29,33,43,45)260 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 26766);