Finite Group DatabasebetaFinite Group Databasebeta| order | 1600 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #360 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C25 · Schur multiplier (H2) = C2 |
| direct product | C25×#360 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C25 | #102 |
| commutator | C8 | C2×C4×C25 |
| Frattini, cosocle | C2×C5×C8 | C22×C5 |
| socle | C2×C5 | C5×#102 |
| prime radical | C5×#135 | C2×C5 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C25 ◁ C2×C4×C25 ◁ C2×C8×C25 ◁ G | |
87 conjugacy classes of subgroups.
| minimal degree | μ(G) = 41 |
| transitive degree | μt(G) = 400 |
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)(27,31,32,40)(28,34)(29,37)(33,35,38,41)(36,39);(26,27)(28,35)(29,38)(30,32)(31,39)(33,37)(34,40)(36,41)400 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1600, 165);