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×#13102 |
| 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 | |
688 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)(6,7,11,14,12,17,10,16)(8,18,22,34,23,36,21,35)(9,24,27,29,26,15,19,32)(13,28,30,25,31,37,33,20);(6,8,19,37)(7,13,29,35)(9,25,10,22)(11,21,26,20)(12,23,27,28)(14,30,24,36)(15,18,16,33)(17,31,32,34);(7,14)(9,26)(10,11)(13,30)(15,32)(16,17)(18,34)(20,25)(21,22)(24,29)(31,33)(35,36)170 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 6090);