Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33359 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C5 · Schur multiplier (H2) = C26 |
| direct product | C5×#33371 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C5 | C22×#99 |
| commutator | C2×C4 | C23×C4×C5 |
| Frattini, cosocle | C2×Q8 | C24×C5 |
| prime radical | C5×#3500 | C2 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C5 ◁ C2×C5×Q8 ◁ C2×C5×H(5,𝔽2) ◁ G | |
1208 conjugacy classes of subgroups.
| minimal degree | μ(G) = 37 |
| transitive degree | μt(G) = 160 |
A minimal faithful permutation representation — 4 generators on 37 points:
(1,2,3,4,5)(6,7,12,17,13,18,9,15)(8,19,20,35,22,30,27,37)(10,16,24,34,11,25,23,33)(14,28,31,26,32,36,29,21);(6,8)(7,14)(9,20)(10,21)(11,26)(12,27)(13,22)(15,29)(16,30)(17,31)(18,32)(19,25)(23,28)(24,36)(33,35)(34,37);(6,9,13,12)(7,15,18,17)(8,20,22,27)(10,24,11,23)(14,29,32,31)(16,34,25,33)(19,35,30,37)(21,36,26,28);(6,10,13,11)(7,16,18,25)(8,21,22,26)(9,23,12,24)(14,30,32,19)(15,33,17,34)(20,28,27,36)(29,35,31,37)230 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 26359);