Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #99 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C16 · Schur multiplier (H2) = C23×C4 |
| direct product | C16×#99 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C16 | C22⋊C4 |
| commutator | C22 | C2×C4×C16 |
| Frattini, cosocle | C23×C8 | C23 |
| socle | C22 | C8×C22⋊C4 |
| prime radical | C22×D4 | C2×C8 |
| LCS | G ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C16 ◁ C23×C16 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
408 conjugacy classes of subgroups; only the 152 normal subgroups are shown.
This group has 408 conjugacy classes of subgroups — too many to draw. Only its 152 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) = 128 |
A minimal faithful permutation representation — 3 generators on 24 points:
(9,10,11,14,12,15,17,20,13,16,18,21,19,22,23,24);(1,2)(3,7,8,5)(4,6);(1,3)(2,5)(4,8)(6,7)176 conjugacy classes — view the full character table.
| maximal subgroup of | C16×B(4,𝔽2), #99×AGL(1,𝔽17) |
| maximal normal subgroup of | C16×B(4,𝔽2) |
| maximal quotient of | #99×AGL(1,𝔽17) |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 35639);