Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13175 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C5 · Schur multiplier (H2) = C22 |
| direct product | C5×#13175 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C5 | #1943 |
| commutator | C2×C8 | C22×C4×C5 |
| Frattini, cosocle | C22×C8 | C23×C5 |
| prime radical | C23×C5 | #137 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C5 ◁ C22×C4×C5 ◁ C5×#405 ◁ G | |
256 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)(7,13)(8,17,20,33)(9,11)(10,24)(14,28,30,31)(15,25)(16,22)(18,29,35,37)(19,34,21,32)(23,26);(6,7)(8,18,20,35)(9,22)(10,25)(11,13)(12,16)(14,21,30,19)(15,26)(17,31,33,28)(23,27)(24,36)(29,32,37,34);(6,8,10,17,11,21,24,34,12,20,26,33,9,19,23,32)(7,14,15,29,13,18,27,28,16,30,25,37,22,35,36,31)170 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 6163);