Finite Group DatabasebetaFinite Group Databasebeta| order | 1600 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D8 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C25 · Schur multiplier (H2) = C22 |
| direct product | C2×C25×#108 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C25 | D4 |
| commutator | C4 | C22×C4×C25 |
| Frattini, cosocle | C2×C4×C5 | C23×C5 |
| socle | C22×C5 | C5×C4⋊C4 |
| prime radical | C23×C5 | C5×Q8 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C25 ◁ C22×C4×C25 ◁ G | |
159 conjugacy classes of subgroups.
| minimal degree | μ(G) = 43 |
| transitive degree | μt(G) = 800 |
A minimal faithful permutation representation — 3 generators on 43 points:
(1,2)(19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43);(3,4,7,10,8,11,6,16)(5,12,14,17,15,18,13,9);(3,5,6,13,8,15,7,14)(4,9,10,12,11,17,16,18)700 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1600, 930);