Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #1408 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C24×C4 |
| direct product | C2×#7498 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #353 |
| commutator | C42 | C2×C42 |
| Frattini, cosocle | C2×#127 | C23 |
| prime radical | C22×#127 | C22 |
| LCS | G ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24 ◁ C23×C4 ◁ C22×#127 ◁ G | |
| LECS | G ▷ C2×#127 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
906 conjugacy classes of subgroups; only the 138 normal subgroups are shown.
This group has 906 conjugacy classes of subgroups — too many to draw. Only its 138 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 22 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 22 points:
(1,2);(7,8)(9,17,18,13)(10,14,11,15)(12,16)(19,22,20,21);(3,4,5,6)(7,9,10,19,12,20,11,18)(8,13,15,21,16,22,14,17)68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 59029);