Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #353 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C211 |
| direct product | C23×#354 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | #99 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C2×D4 | C25 |
| prime radical | C24×D4 | C22 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C25 ◁ C24×D4 ◁ G | |
| LECS | G ▷ C2×D4 ▷ C22 ▷ C2 ▷ C1 | |
5306 conjugacy classes of subgroups; only the 1090 normal subgroups are shown.
This group has 5306 conjugacy classes of subgroups — too many to draw. Only its 1090 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)(9,15,17,22)(10,18,20,13)(11,14)(12,16)(19,21);(7,9,11,17)(8,12,14,21)(10,19,20,16)(13,22,18,15)104 conjugacy classes — view the full character table.
| maximal subgroup of | C23×#2127, C22×#13348, C23×#2197, … |
| maximal normal subgroup of | C23×#2127, C22×#13348, C23×#2197, … |
| maximal quotient of | #72263636, #72318420 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7529652);