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) = C22×C4 |
| direct product | C16×#101 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C16 | C22⋊C4 |
| commutator | C22 | C2×C4×C16 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C22 | C8×C22⋊C4 |
| prime radical | C23 | C2×C4×C8 |
| LCS | G ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C16 ◁ C2×C4×C16 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
306 conjugacy classes of subgroups; only the 152 normal subgroups are shown.
This group has 306 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) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 3 generators on 32 points:
(17,18,19,22,20,23,25,28,21,24,26,29,27,30,31,32);(1,2,4,8,5,7,13,15)(3,9,10,16,11,6,12,14);(1,3,5,11)(2,6,7,9)(4,12,13,10)(8,16,15,14)176 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 35703);