Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11858 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C5 · Schur multiplier (H2) = C23 |
| direct product | C5×#11894 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | C2×#102 |
| commutator | C2×C8 | C22×C4×C5 |
| Frattini, cosocle | C22×C8 | C23×C5 |
| prime radical | C5×#3050 | C2 |
| LCS | G ▷ C2×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C23×C4×C5 ◁ C23×C5×C8 ◁ G | |
394 conjugacy classes of subgroups.
| minimal degree | μ(G) = 45 |
| transitive degree | μt(G) = 320 |
A minimal faithful permutation representation — 3 generators on 45 points:
(1,2,3,4,5)(6,7)(8,12,13,10)(9,11)(14,15,19,25,21,27,20,26)(16,28,31,43,33,44,32,39)(17,34,18,37,36,24,35,23)(22,38,40,30,42,29,41,45);(6,8)(7,10)(9,13)(11,12)(14,16)(15,22)(17,30)(18,38)(19,32)(20,31)(21,33)(23,28)(24,43)(25,41)(26,40)(27,42)(29,35)(34,39)(36,45)(37,44);(8,13)(10,12)(14,17,20,35,21,36,19,18)(15,23,25,34,27,37,26,24)(16,29,32,30,33,38,31,45)(22,39,40,28,42,43,41,44)200 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 4882);