Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13096 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C5 · Schur multiplier (H2) = C24 |
| direct product | C5×#13099 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C5 | #1898 |
| commutator | C22×C4 | C22×C4×C5 |
| Frattini, cosocle | C2×#41 | C23×C5 |
| prime radical | C5×#3008 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C5 ◁ C2×C4×C5 ◁ C2×C5×H(5,𝔽2) ◁ G | |
666 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)(6,7,11,16)(8,17)(9,23)(10,15,27,32)(12,19,30,20)(13,26)(14,18)(21,34)(22,35)(24,31)(25,28)(29,36,33,37);(6,8,18,20,11,22,9,19)(7,12,23,35,16,30,14,17)(10,25,26,36,27,21,24,37)(13,28,15,33,31,34,32,29);(6,9,11,18)(7,13,16,31)(8,19,22,20)(10,26,27,24)(12,28,30,34)(14,32,23,15)(17,29,35,33)(21,37,25,36)170 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 6087);