Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #99 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C8 · Schur multiplier (H2) = C26×C4 |
| direct product | C2×C8×#99 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C8 | C22⋊C4 |
| commutator | C22 | C22×C4×C8 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C23 | C4×C22⋊C4 |
| prime radical | C23×D4 | C2×C4 |
| LCS | G ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C8 ◁ C24×C8 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
1686 conjugacy classes of subgroups.
| minimal degree | μ(G) = 18 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 18 points:
(1,2);(3,4,5,7,6,8,9,10);(11,12)(13,17,18,15)(14,16);(11,13)(12,15)(14,18)(16,17)176 conjugacy classes — view the full character table.
| maximal subgroup of | #99×#106, #72313490, C2×#301866, … |
| maximal normal subgroup of | #99×#106, #72313490, C2×#301866, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 89256);