Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #60457 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C5 · Schur multiplier (H2) = C27×C4 |
| direct product | C5×#60622 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | C23×D4 |
| commutator, Frattini, cosocle | C2×C4 | C25×C5 |
| prime radical | C5×#3498 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C4×C5×#41 ◁ G | |
2178 conjugacy classes of subgroups.
| minimal degree | μ(G) = 29 |
| transitive degree | μt(G) = 320 |
A minimal faithful permutation representation — 5 generators on 29 points:
(1,2,3,4,5)(6,7)(8,11)(9,10)(12,13)(15,21)(16,23)(18,19)(22,28)(24,29)(26,27);(7,10)(11,13)(14,15,19,16,20,23,18,21)(17,22,27,24,25,29,26,28);(6,8,9,12)(7,11,10,13)(14,16,18,15,20,21,19,23)(17,24,26,22,25,28,27,29);(14,17)(15,22)(16,24)(18,26)(19,27)(20,25)(21,28)(23,29);(17,25)(22,29)(24,28)(26,27)320 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 53610);