Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23886 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C24 |
| direct product | C5×#26404 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | D42 |
| commutator, Frattini, cosocle | C22×C4 | C24×C5 |
| prime radical | C2×C5×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C22×C4×C5 ◁ G | |
706 conjugacy classes of subgroups.
| minimal degree | μ(G) = 45 |
| transitive degree | μt(G) = 320 |
A minimal faithful permutation representation — 4 generators on 45 points:
(1,2,3,4,5)(6,7,10,12)(8,11,9,13)(14,15)(16,26)(17,21)(18,24)(19,25)(20,29)(22,31)(23,35)(27,40)(28,37)(30,38)(32,41)(33,42)(34,39)(36,43)(44,45);(6,8,10,9)(7,11,12,13)(14,16,19,30)(15,20,25,39)(17,28,33,43)(18,34,35,29)(21,40,42,45)(22,37,41,36)(23,38,24,26)(27,31,44,32);(6,9,10,8)(7,11,12,13)(14,17)(15,21)(16,27)(18,31)(19,33)(20,36)(22,24)(23,41)(25,42)(26,40)(28,34)(29,43)(30,44)(32,35)(37,39)(38,45);(14,17)(15,22)(16,28)(18,32)(19,33)(20,37)(21,24)(23,42)(25,41)(26,40)(27,34)(29,44)(30,43)(31,35)(36,39)(38,45)215 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 19392);