Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #18252 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C25 |
| direct product | C5×#18262 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | C2×#120 |
| commutator, Frattini, cosocle | C22×C4 | C24×C5 |
| prime radical | C5×#3534 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C23×C4×C5 ◁ G | |
1012 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)(6,7,11,9)(8,16,20,32)(10,23,25,35)(12,14,27,21)(13,28,17,18)(15,30,22,26)(19,31,34,36)(24,29,37,33);(6,8)(7,13)(9,17)(10,24)(11,20)(12,19)(14,29)(15,31)(16,23)(18,26)(21,33)(22,36)(25,37)(27,34)(28,30)(32,35);(6,9)(7,11)(8,17)(10,15)(12,21)(13,20)(14,27)(16,18)(19,33)(22,25)(23,26)(24,31)(28,32)(29,34)(30,35)(36,37);(7,14)(9,21)(13,29)(15,23)(16,31)(17,33)(22,35)(32,36)200 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 11250);