Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33522 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C24 |
| direct product | C5×#33634 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | C22×D8 |
| commutator, Frattini, cosocle | C2×C8 | C24×C5 |
| prime radical | C2×C5×#135 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C5×#117 ◁ C5×#405 ◁ G | |
534 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)(6,7,9,11)(8,12,13,10)(14,15,20,25)(16,21,28,39)(17,23,19,35)(18,31,34,45)(22,40,42,27)(24,30,43,33)(26,36,29,38)(32,37,44,41);(6,8,9,13)(7,10,11,12)(14,16)(15,21)(17,29)(18,27)(19,26)(20,28)(22,31)(23,38)(24,37)(25,39)(30,44)(32,33)(34,40)(35,36)(41,43)(42,45);(14,16)(15,22)(17,26)(18,32)(19,29)(20,28)(21,31)(23,37)(24,38)(25,42)(27,33)(30,40)(34,44)(35,41)(36,43)(39,45);(14,17,20,19)(15,23,25,35)(16,26,28,29)(18,30,34,33)(21,36,39,38)(22,37,42,41)(24,31,43,45)(27,32,40,44)230 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 26622);