Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #1339 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
| direct product | C22×#1341 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #327 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C2×M16 | C24 |
| prime radical | C22×#469 | C2 |
| LCS | G ▷ C2×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ C23×M16 ◁ G | |
| LECS | G ▷ C2×M16 ▷ C2×C4 ▷ C2 ▷ C1 | |
1370 conjugacy classes of subgroups; only the 282 normal subgroups are shown.
This group has 1370 conjugacy classes of subgroups — too many to draw. Only its 282 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 20 points:
(1,2);(3,4);(5,6,8,13,10,14,9,18)(7,11,15,12,17,20,16,19);(6,11)(7,15)(8,9)(12,13)(14,20)(16,17)(18,19)80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 388636);