Finite Group DatabasebetaFinite Group Databasebeta| order | 1664 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #509 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C13 · Schur multiplier (H2) = C25 |
| direct product | C2×C13×#509 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C13 | C2×D8 |
| commutator, Frattini, cosocle | C8 | C24×C13 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C13 ◁ C22×C4×C13 ◁ C22×C8×C13 ◁ G | |
400 conjugacy classes of subgroups; only the 184 normal subgroups are shown.
This group has 400 conjugacy classes of subgroups — too many to draw. Only its 184 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 31 |
| transitive degree | μt(G) = 416 |
A minimal faithful permutation representation — 4 generators on 31 points:
(1,2)(19,20,21,22,23,24,25,26,27,28,29,30,31);(3,4)(5,8)(6,15)(7,9)(10,12)(11,13)(14,16)(17,18);(4,8)(5,12)(6,13)(9,16)(10,11)(14,18)(15,17);(4,9)(8,16)(10,17)(11,15)416 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1664, 2144);