Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of AGL(1,ℤ/8) |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C8 · Schur multiplier (H2) = C29 |
| direct product | C22×#1565 |
| subgroup | G/H | |
|---|---|---|
| center | C23×C4 | C2×D4 |
| commutator | C4 | C24×C8 |
| Frattini, cosocle | C42 | C25 |
| socle | C24 | C2×C4⋊C4 |
| prime radical | C25 | C4⋊C4 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C4 ◁ C23×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
2514 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 20 points:
(1,2);(3,4);(6,9)(7,11)(10,12)(13,14,15,17,16,18,19,20);(5,6,7,9,8,10,11,12);(6,10)(9,12)176 conjugacy classes — view the full character table.
| maximal subgroup of | C22×C8×AGL(1,ℤ/8) |
| maximal normal subgroup of | C22×C8×AGL(1,ℤ/8) |
| maximal quotient of | C22×#8830 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6251501);