Finite Group DatabasebetaFinite Group Databasebeta| order | 1600 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #453 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C25 · Schur multiplier (H2) = C23 |
| direct product | C25×#453 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C25 | #120 |
| commutator | C2×C4 | C23×C25 |
| Frattini, cosocle | C2×C4×C5 | C23×C5 |
| socle | C2×C5 | C5×#120 |
| prime radical | C5×#453 | C5 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C25 ◁ C2×C4×C25 ◁ G | |
252 conjugacy classes of subgroups; only the 81 normal subgroups are shown.
This group has 252 conjugacy classes of subgroups — too many to draw. Only its 81 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 33 |
| transitive degree | μt(G) = 200 |
A minimal faithful permutation representation — 3 generators on 33 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)(28,32)(31,33);(26,27)(28,33)(29,31)(30,32);(28,32)(30,33)400 conjugacy classes — view the full character table.
| Fitting subgroup of | D25×#453, #453×#1035 |
| maximal subgroup of | D25×#453, #453×5−1+2 |
| maximal normal subgroup of | D25×#453, #453×5−1+2 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1600, 954);