Finite Group DatabasebetaFinite Group Databasebeta| order | 1600 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #327 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C25 · Schur multiplier (H2) = C1 |
| direct product | C25×#330 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C25 | C22⋊C4 |
| commutator | C2×C4 | C2×C4×C25 |
| Frattini, cosocle | C42×C5 | C22×C5 |
| socle, prime radical | C22×C5 | C5×C22⋊C4 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C25 ◁ C42×C25 ◁ G | |
90 conjugacy classes of subgroups.
| minimal degree | μ(G) = 41 |
| transitive degree | μt(G) = 1600 |
A minimal faithful permutation representation — 2 generators on 41 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)(27,30,31,32)(34,35,38,40,39,36,37,41);(26,27)(28,32)(29,31)(30,33)(34,36,39,35)(37,40,38,41)475 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1600, 135);