Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12767 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C5 · Schur multiplier (H2) = C25 |
| direct product | C5×#12770 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C5 | #1790 |
| commutator | C2×D4 | C22×C4×C5 |
| Frattini, cosocle | C2×#41 | C23×C5 |
| prime radical | C5×#3551 | C2 |
| LCS | G ▷ C2×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C5 ◁ C2×C4×C5 ◁ C2×C5×#41 ◁ G | |
752 conjugacy classes of subgroups.
| minimal degree | μ(G) = 37 |
| transitive degree | μt(G) = 160 |
A minimal faithful permutation representation — 3 generators on 37 points:
(1,2,3,4,5)(6,7,10,15,11,13,23,28)(8,16,19,32,18,33,36,37)(9,20,22,31,21,14,24,29)(12,25,27,17,26,34,30,35);(6,8,11,18)(7,12,13,26)(9,17,21,35)(10,19,23,36)(14,16,20,33)(15,30,28,27)(22,34,24,25)(29,37,31,32);(7,13)(10,23)(12,26)(19,36)(22,24)(25,34)(29,31)(32,37)170 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 5758);