Finite Group DatabasebetaFinite Group Databasebeta| order | 1600 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #360 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C52 · Schur multiplier (H2) = C2×C5 |
| direct product | C52×#360 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C52 | #102 |
| commutator | C8 | C2×C4×C52 |
| Frattini, cosocle | C2×C8 | C22×C52 |
| prime radical | C52×#135 | C2 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C52 ◁ C2×C4×C52 ◁ C2×C52×C8 ◁ G | |
232 conjugacy classes of subgroups.
| 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)(13,19)(14,22)(18,20,23,26)(21,24);(6,7,8,9,10)(11,12)(13,20)(14,23)(15,17)(16,24)(18,22)(19,25)(21,26)400 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1600, 1993);