Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #18252 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C26 |
| direct product | C5×#18261 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | C2×#120 |
| commutator, Frattini, cosocle | C22×C4 | C24×C5 |
| prime radical | C5×#3536 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C23×C4×C5 ◁ G | |
1192 conjugacy classes of subgroups.
| minimal degree | μ(G) = 29 |
| transitive degree | μt(G) = 320 |
A minimal faithful permutation representation — 4 generators on 29 points:
(1,2,3,4,5)(10,13)(14,15,20,25)(16,26,17,22)(18,24,19,23)(21,28,29,27);(6,7)(8,10)(9,11)(12,13)(14,16,20,17)(15,21,25,29)(18,27,19,28)(22,23,26,24);(6,8)(7,10)(9,12)(11,13)(14,17,20,16)(15,22,25,26)(18,27,19,28)(21,23,29,24);(14,18,20,19)(15,23,25,24)(16,27,17,28)(21,26,29,22)200 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 11249);