Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #1406 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C23×C4 |
| direct product | C2×#7508 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #353 |
| commutator | C42 | C2×C42 |
| Frattini, cosocle | C2×#124 | C23 |
| prime radical | C23×D4 | D4 |
| LCS | G ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24 ◁ C23×C4 ◁ C22×#124 ◁ G | |
| LECS | G ▷ C2×#124 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
726 conjugacy classes of subgroups; only the 138 normal subgroups are shown.
This group has 726 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) = 128 |
A minimal faithful permutation representation — 3 generators on 22 points:
(1,2);(7,8,9,15)(10,16,20,13)(11,14,19,21)(12,17,18,22);(3,4,5,6)(7,9,10,19,12,18,11,20)(8,13,16,15,17,21,14,22)56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 59061);