Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #18132 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C25 |
| direct product | C5×#18138 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | C2×#120 |
| commutator, Frattini, cosocle | C22×C4 | C24×C5 |
| prime radical | C25×C5 | D4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C23×C4×C5 ◁ G | |
1054 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)(6,7,11,15)(8,16,18,32)(9,20,22,34)(10,13,24,28)(12,25,27,17)(14,29,31,23)(19,30,33,37)(21,26,35,36);(6,8,10,19)(7,12,13,26)(9,21,23,17)(11,18,24,33)(14,30,20,16)(15,27,28,36)(22,35,29,25)(31,37,34,32);(9,22)(14,31)(16,32)(17,25)(20,34)(21,35)(23,29)(30,37);(7,13)(12,26)(14,20)(15,28)(16,30)(27,36)(31,34)(32,37)185 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 11126);