Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #29736 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C22×C42 |
| direct product | C5×#29738 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C5 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24×C5 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C5 ◁ C22×C5×#41 ◁ G | |
1082 conjugacy classes of subgroups.
| minimal degree | μ(G) = 33 |
| transitive degree | μt(G) = 640 |
A minimal faithful permutation representation — 4 generators on 33 points:
(1,2,3,4,5)(10,11)(12,15)(13,14)(16,17)(18,19,23,27,22,26,21,25)(20,28,29,32,30,24,31,33);(6,7)(8,9)(11,14)(15,17)(18,20)(19,24)(21,31)(22,30)(23,29)(25,32)(26,28)(27,33);(7,9)(10,12)(11,15)(13,16)(14,17)(19,25)(21,23)(24,32)(26,27)(28,33)(29,31);(12,16)(15,17)(18,21,22,23)(19,25,26,27)(20,29,30,31)(24,33,28,32)200 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 22726);