Finite Group DatabasebetaFinite Group Databasebeta| order | 1600 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42×C52 · Schur multiplier (H2) = C2×C42×C5 |
| direct product | C52×#376 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4×C52 | C22 |
| commutator | C2 | C2×C42×C52 |
| Frattini, cosocle | C23 | C23×C52 |
| socle, prime radical | C23×C52 | C23 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4×C52 ◁ G | |
712 conjugacy classes of subgroups.
| minimal degree | μ(G) = 26 |
| transitive degree | μt(G) = 1600 |
A minimal faithful permutation representation — 3 generators on 26 points:
(1,2,3,4,5)(11,12,13,14)(19,20)(21,24)(22,23)(25,26);(6,7,8,9,10)(15,16,17,18)(20,23)(24,26);(19,21,22,25)(20,24,23,26)1000 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1600, 5465);