Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #471 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C5×C8 · Schur multiplier (H2) = C22 |
| direct product | C5×#13164 |
| subgroup | G/H | |
|---|---|---|
| center | C5×C8 | #121 |
| commutator | C2×C4 | C22×C5×C8 |
| Frattini, cosocle | C22×C8 | C23×C5 |
| socle | C2×C5 | #1939 |
| prime radical | C23×C5 | #131 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C5×C8 ◁ C22×C5×C8 ◁ G | |
284 conjugacy classes of subgroups.
| minimal degree | μ(G) = 37 |
| transitive degree | μt(G) = 160 |
A minimal faithful permutation representation — 3 generators on 37 points:
(1,2,3,4,5)(6,7,11,17,12,16,27,34,13,18,26,15,10,28,9,25)(8,19,23,36,21,14,29,32,24,31,37,35,22,33,20,30);(6,8,11,23,12,21,27,29,13,24,26,37,10,22,9,20)(7,14,17,32,16,31,34,35,18,33,15,30,28,19,25,36);(6,9,10,26,13,27,12,11)(7,15,16,25,18,17,28,34)(8,20,22,37,24,29,21,23)(14,30,31,36,33,32,19,35)320 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 6152);