Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #120 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C16 · Schur multiplier (H2) = C25×C4 |
| direct product | C16×#127 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C16 | C23 |
| commutator | C22 | C23×C16 |
| Frattini, cosocle | C22×C8 | C24 |
| socle | C23 | C23×C8 |
| prime radical | C2×#127 | C8 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C16 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
888 conjugacy classes of subgroups; only the 342 normal subgroups are shown.
This group has 888 conjugacy classes of subgroups — too many to draw. Only its 342 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 24 points:
(9,10,11,14,12,15,17,20,13,16,18,21,19,22,23,24);(2,4)(6,8);(1,2,3,4);(5,6,7,8)224 conjugacy classes — view the full character table.
| maximal subgroup of | #127×AGL(1,𝔽17), C16×#453 |
| maximal normal subgroup of | C16×#453 |
| maximal quotient of | #127×AGL(1,𝔽17) |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 403281);