Finite Group DatabasebetaFinite Group Databasebeta| order | 1600 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C52×C8 · Schur multiplier (H2) = C22×C5 |
| direct product | C52×C8×Q8 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C52×C8 | C22 |
| commutator | C2 | C22×C52×C8 |
| Frattini, cosocle | C2×C4 | C23×C52 |
| socle, prime radical | C22×C52 | C22×C4 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C52×C8 ◁ G | |
408 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)(11,12,13,14,15,16,17,18);(6,7,8,9,10)(19,20,21,22)(23,26,25,24);(19,23,21,25)(20,24,22,26)1000 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1600, 5534);