Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23400 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C27 |
| direct product | C5×#23403 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | C2×#120 |
| commutator, Frattini, cosocle | C22×C4 | C24×C5 |
| prime radical | C2×C5×#575 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C23×C4×C5 ◁ G | |
1942 conjugacy classes of subgroups.
| minimal degree | μ(G) = 41 |
| transitive degree | μt(G) = 320 |
A minimal faithful permutation representation — 4 generators on 41 points:
(1,2,3,4,5)(6,7)(8,9)(10,11)(12,21)(13,18)(14,19)(15,28)(16,20)(17,23)(22,37)(24,31)(25,34)(26,36)(27,35)(29,32)(30,40)(33,38)(39,41);(10,12)(11,17)(13,22)(14,23)(15,29)(16,24)(18,33)(19,31)(20,32)(21,28)(25,38)(26,40)(27,39)(30,35)(34,41)(36,37);(7,9)(10,13,16,27)(11,18,20,35)(12,22,24,39)(14,26,15,25)(17,30,32,33)(19,36,28,34)(21,37,31,41)(23,40,29,38);(10,14,16,15)(11,19,20,28)(12,23,24,29)(13,25,27,26)(17,31,32,21)(18,34,35,36)(22,38,39,40)(30,37,33,41)230 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 16391);