Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23400 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C26 |
| direct product | C5×#23464 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | C2×#120 |
| commutator, Frattini, cosocle | C22×C4 | C24×C5 |
| prime radical | C24×C5 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C23×C4×C5 ◁ G | |
1146 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)(6,7,9,11)(8,12,13,10)(14,15)(16,21)(17,28)(18,23)(19,32)(20,24)(22,30)(25,41)(26,35)(27,37)(29,39)(31,40)(33,36)(34,42)(38,44)(43,45);(6,8,9,13)(7,10,11,12)(14,16)(15,21)(17,25)(18,26)(19,33)(20,27)(22,38)(23,35)(24,37)(28,41)(29,42)(30,44)(31,43)(32,36)(34,39)(40,45);(14,17,20,31)(15,22,24,39)(16,25,27,43)(18,30,19,29)(21,34,37,38)(23,40,32,28)(26,44,33,42)(35,41,36,45);(14,18,20,19)(15,23,24,32)(16,26,27,33)(17,29,31,30)(21,35,37,36)(22,28,39,40)(25,42,43,44)(34,45,38,41)230 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 16452);