Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #360 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C210 |
| direct product | C23×#361 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | #102 |
| commutator | C8 | C24×C4 |
| Frattini, cosocle | C2×C8 | C25 |
| prime radical | C24×D8 | C2 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C24×C4 ◁ C24×C8 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
5074 conjugacy classes of subgroups; only the 1346 normal subgroups are shown.
This group has 5074 conjugacy classes of subgroups — too many to draw. Only its 1346 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 — 5 generators on 22 points:
(1,2);(3,4);(5,6);(7,8,10,14,11,16,21,15)(9,17,19,13,20,12,18,22);(8,12)(9,18)(10,21)(13,14)(15,22)(16,17)(19,20)128 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#147868, C22×#13331, C22×#12647, … |
| maximal normal subgroup of | C2×#147868, C22×#13331, C22×#12647, … |
| maximal quotient of | C2×#481279, C2×#494044, C22×#12736, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7529704);