Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of AΣL(1,𝔽16) |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C8 · Schur multiplier (H2) = C26 |
| direct product | C22×#1320 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | #99 |
| commutator | C23 | C23×C8 |
| Frattini, cosocle | C2×M16 | C24 |
| socle | C23 | #323 |
| prime radical | C26 | C8 |
| LCS | G ▷ C23 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C23×C4 ◁ C23×M16 ◁ G | |
| LECS | G ▷ C2×M16 ▷ C2×C4 ▷ C2 ▷ C1 | |
2090 conjugacy classes of subgroups; only the 322 normal subgroups are shown.
This group has 2090 conjugacy classes of subgroups — too many to draw. Only its 322 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);(5,6,7,11,8,12,14,18,9,10,13,16,15,17,19,20);(6,10)(12,17)104 conjugacy classes — view the full character table.
| maximal subgroup of | #72213784, #72215054, #72216373, … |
| maximal normal subgroup of | #72213784, #72215054, #72216373, … |
| maximal quotient of | C22×#7105, C2×#129224, #72293302 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 387200);