Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #18524 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C25 |
| direct product | C5×#18543 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | C2×#120 |
| commutator, Frattini, cosocle | C42 | C24×C5 |
| prime radical | C5×#3482 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C2×C42×C5 ◁ G | |
954 conjugacy classes of subgroups.
| minimal degree | μ(G) = 29 |
| transitive degree | μt(G) = 640 |
A minimal faithful permutation representation — 4 generators on 29 points:
(1,2,3,4,5)(7,11)(8,9)(12,13)(14,15,20,25)(16,22,27,28)(17,21,26,29)(18,24,19,23);(6,7)(8,11)(9,13)(10,12)(14,16,20,27)(15,21,25,29)(17,18,26,19)(22,23,28,24);(14,17)(15,22)(16,18)(19,27)(20,26)(21,23)(24,29)(25,28);(6,8,10,9)(7,11,12,13)(14,18,20,19)(15,23,25,24)(16,26,27,17)(21,28,29,22)200 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 11531);