Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #29748 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C23×C4 |
| direct product | C5×#29749 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | C2×#121 |
| commutator, Frattini, cosocle | C22×C4 | C24×C5 |
| prime radical | C22×C5×#41 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C23×C4×C5 ◁ G | |
904 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)(8,11)(9,10)(12,13)(14,15,18,23,20,25,19,24)(16,21,27,36,28,37,34,41)(17,29,31,40,33,39,32,22)(26,38,30,44,42,45,43,35);(7,10)(11,13)(14,16)(15,21)(17,30)(18,27)(19,34)(20,28)(22,38)(23,36)(24,41)(25,37)(26,32)(29,44)(31,42)(33,43)(35,39)(40,45);(6,8)(7,11)(9,12)(10,13)(14,17,20,33)(15,22,25,40)(16,26,28,42)(18,32,19,31)(21,35,37,44)(23,39,24,29)(27,43,34,30)(36,45,41,38);(8,12)(11,13)200 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 22737);