Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of AΣL(1,𝔽16) |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C16 · Schur multiplier (H2) = C23 |
| direct product | C2×#7336 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C8 | #99 |
| commutator | C23 | C22×C16 |
| Frattini, cosocle | C2×M32 | C23 |
| socle | C22 | #1314 |
| prime radical | C25 | C16 |
| LCS | G ▷ C23 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C8 ◁ C22×C8 ◁ C22×M32 ◁ G | |
| LECS | G ▷ C2×M32 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
426 conjugacy classes of subgroups; only the 98 normal subgroups are shown.
This group has 426 conjugacy classes of subgroups — too many to draw. Only its 98 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 34 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 34 points:
(1,2);(3,4,5,10,6,11,14,22,7,12,15,23,16,24,27,32,8,9,13,19,17,20,25,29,18,21,26,30,28,31,33,34);(4,9)(11,20)(12,21)(24,31)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 56719);