Finite Group DatabasebetaFinite Group Databasebeta| order | 1600 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #120 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C52 · Schur multiplier (H2) = C23×C5 |
| direct product | C52×#383 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23×C52 | C23 |
| commutator | C22 | C22×C4×C52 |
| Frattini, cosocle | C23 | C23×C52 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23×C52 ◁ G | |
552 conjugacy classes of subgroups.
| minimal degree | μ(G) = 30 |
| transitive degree | μt(G) = 1600 |
A minimal faithful permutation representation — 3 generators on 30 points:
(1,2,3,4,5)(11,12,13,14)(17,21)(19,22)(24,27)(25,29);(6,7,8,9,10)(15,16,18,20)(17,19,21,22)(23,24)(25,28)(26,27)(29,30);(15,17)(16,19)(18,21)(20,22)(23,25,26,29)(24,28,27,30)700 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1600, 5472);