Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #99 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C16 · Schur multiplier (H2) = C27 |
| direct product | C22×#1322 |
| subgroup | G/H | |
|---|---|---|
| center | C23×C4 | C22⋊C4 |
| commutator | C22 | C23×C16 |
| Frattini, cosocle | C22×C8 | C24 |
| socle | C24 | C22⋊C8 |
| prime radical | C25 | C2×C8 |
| LCS | G ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C4 ◁ C24×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
1314 conjugacy classes of subgroups; only the 442 normal subgroups are shown.
This group has 1314 conjugacy classes of subgroups — too many to draw. Only its 442 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 28 points:
(1,2);(3,4);(5,6)(7,11,12,9)(8,10)(13,14,15,18,16,19,21,24,17,20,22,25,23,26,27,28);(5,7,8,12)(6,9,10,11)176 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 387216);