Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33522 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C26 |
| direct product | C5×#33527 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | C22×D8 |
| commutator, Frattini, cosocle | C2×C8 | C24×C5 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C2×C5×#41 ◁ C2×C5×#131 ◁ G | |
1098 conjugacy classes of subgroups.
| minimal degree | μ(G) = 41 |
| transitive degree | μt(G) = 640 |
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)(14,27)(15,25)(16,20)(17,23)(19,29)(22,38)(24,39)(26,36)(28,32)(30,37)(31,35)(33,41)(34,40);(10,12,16,24)(11,17,20,34)(13,22,15,31)(14,28,30,41)(18,32,25,33)(19,35,36,38)(21,37,39,27)(23,26,40,29);(7,9)(10,11)(12,17)(13,25)(14,19)(15,18)(16,20)(21,23)(22,33)(24,34)(26,37)(27,29)(28,35)(30,36)(31,32)(38,41)(39,40);(10,13,16,15)(11,18,20,25)(12,22,24,31)(14,26,30,29)(17,32,34,33)(19,27,36,37)(21,38,39,35)(23,28,40,41)230 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 26515);