Finite Group DatabasebetaFinite Group Databasebeta| order | 1600 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D8 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C52 · Schur multiplier (H2) = C24×C5 |
| direct product | C2×C52×#102 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C52 | D4 |
| commutator | C4 | C22×C4×C52 |
| Frattini, cosocle | C2×C4 | C23×C52 |
| prime radical | C22×C52×D4 | C2 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C52 ◁ C22×C4×C52 ◁ G | |
808 conjugacy classes of subgroups; only the 392 normal subgroups are shown.
This group has 808 conjugacy classes of subgroups — too many to draw. Only its 392 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 800 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2)(15,16,17,18,19);(3,4,5,6)(7,8)(9,11)(10,13)(12,14)(20,21,22,23,24);(8,11)(9,14)(12,13)700 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1600, 5503);