Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #25196 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C25 |
| direct product | C5×#25199 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C5 | #2424 |
| commutator, Frattini, cosocle | C22×C4 | C24×C5 |
| prime radical | C5×#3019 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C5 ◁ C22×C4×C5 ◁ G | |
990 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)(6,7,12,16,13,17,11,27)(8,18,20,14,21,31,28,32)(9,22,24,33,10,25,26,15)(19,30,36,37,23,29,35,34);(6,8,12,20,13,21,11,28)(7,14,16,31,17,32,27,18)(9,23,24,35,10,19,26,36)(15,30,22,37,33,29,25,34);(6,9,13,10)(7,15,17,33)(8,19,21,23)(11,24,12,26)(14,29,32,30)(16,25,27,22)(18,34,31,37)(20,35,28,36);(6,10,13,9)(7,15,17,33)(8,19,21,23)(11,24,12,26)(14,30,32,29)(16,22,27,25)(18,34,31,37)(20,36,28,35)200 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 18187);