Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2846 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C29 |
| direct product | C2×#21487 |
| subgroup | G/H | |
|---|---|---|
| center | C23×C4 | C24 |
| commutator | C23 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| socle | C24 | C23×C4 |
| prime radical | C22×#121 | C4 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3466 conjugacy classes of subgroups.
| minimal degree | μ(G) = 18 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 18 points:
(1,2);(4,6)(7,8)(9,10)(11,12,14,13,15,16,18,17);(3,4,5,6);(8,10);(11,13,14,16,15,17,18,12)176 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#6384020, C2×D4×#435, C2×#6372761, … |
| maximal normal subgroup of | C2×#6384020, C2×D4×#435, C2×#6372761, … |
| maximal quotient of | C2×#210752 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6279947);