Finite Group DatabasebetaFinite Group Databasebeta| order | 1600 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #120 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C25 · Schur multiplier (H2) = C24 |
| direct product | C25×#380 |
| subgroup | G/H | |
|---|---|---|
| center | C23×C25 | C23 |
| commutator | C22 | C22×C4×C25 |
| Frattini, socle, cosocle | C23×C5 | C23×C5 |
| prime radical | C24×C5 | C22×C5 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23×C25 ◁ G | |
351 conjugacy classes of subgroups; only the 171 normal subgroups are shown.
This group has 351 conjugacy classes of subgroups — too many to draw. Only its 171 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 37 |
| transitive degree | μt(G) = 800 |
A minimal faithful permutation representation — 3 generators on 37 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25)(27,29)(30,31,32,33)(35,37);(34,35,36,37);(26,27)(28,29)(30,31)(32,33)700 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1600, 881);