Finite Group DatabasebetaFinite Group Databasebeta| order | 1600 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #327 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C52 · Schur multiplier (H2) = C2×C5 |
| direct product | C52×#328 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C52 | C22⋊C4 |
| commutator | C2×C4 | C2×C4×C52 |
| Frattini, cosocle | C22×C4 | C22×C52 |
| prime radical | C23×C52 | D4 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C52 ◁ C22×C4×C52 ◁ G | |
320 conjugacy classes of subgroups; only the 136 normal subgroups are shown.
This group has 320 conjugacy classes of subgroups — too many to draw. Only its 136 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 26 |
| transitive degree | μt(G) = 400 |
A minimal faithful permutation representation — 2 generators on 26 points:
(1,2,3,4,5)(12,16,17,25)(13,20)(18,26,19,21)(22,24);(6,7,8,9,10)(11,12,14,18)(13,21,20,25)(15,19,23,17)(16,24,26,22)475 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1600, 1961);