Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33517 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C24 |
| direct product | C5×#33673 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | C22×D8 |
| commutator, Frattini, cosocle | C2×C8 | C24×C5 |
| prime radical | C23×C5 | C2×D8 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C2×C42×C5 ◁ C2×C4×C5×C8 ◁ G | |
542 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)(8,11)(9,10)(12,13)(14,15,19,23)(16,24,25,43)(17,28,32,42)(18,33,30,22)(20,35,36,26)(21,38,40,31)(27,44,34,39)(29,41,45,37);(7,10)(11,13);(14,16,19,25)(15,20,23,36)(17,29,32,45)(18,34,30,27)(21,39,40,44)(22,41,33,37)(24,42,43,28)(26,38,35,31);(6,8,9,12)(7,11,10,13)(16,25)(20,36)(24,43)(26,35)(27,34)(29,45)(37,41)(39,44)230 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 26661);