Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #120 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C16 · Schur multiplier (H2) = C24 |
| direct product | C2×#13660 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C16 | C23 |
| commutator | C22 | C23×C16 |
| Frattini, cosocle | C2×C16 | C24 |
| socle, prime radical | C23 | C22×C16 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C16 ◁ G | |
| LECS | G ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
342 conjugacy classes of subgroups.
| minimal degree | μ(G) = 42 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 4 generators on 42 points:
(1,2);(3,4)(5,10)(6,8)(7,9)(11,12,14,19,15,20,26,34,16,13,22,23,28,24,37,38,17,21,27,35,29,36,40,42,30,25,18,31,32,39,33,41);(3,5,7,6)(4,8,9,10);(3,6,7,5)(4,8,9,10)(11,13,14,23,15,24,26,38,16,21,22,35,28,36,37,42,17,25,27,31,29,39,40,41,30,12,18,19,32,20,33,34)224 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 419979);