Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #99 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C32 · Schur multiplier (H2) = C24 |
| direct product | C2×#7362 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C8 | C22⋊C4 |
| commutator | C22 | C22×C32 |
| Frattini, cosocle | C22×C16 | C23 |
| socle | C23 | C22⋊C16 |
| prime radical | C24 | C2×C16 |
| LCS | G ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C8 ◁ C23×C16 ◁ G | |
| LECS | G ▷ C22×C16 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
302 conjugacy classes of subgroups; only the 122 normal subgroups are shown.
This group has 302 conjugacy classes of subgroups — too many to draw. Only its 122 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 42 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 3 generators on 42 points:
(1,2);(3,4)(5,9,10,7)(6,8)(11,12,13,17,14,18,21,27,15,19,22,28,24,30,33,37,16,20,23,29,25,31,34,38,26,32,35,39,36,40,41,42);(3,5,6,10)(4,7,8,9)176 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 56795);