Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #392 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C212 |
| direct product | C22×#1444 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C26 | C23 |
| commutator | C23 | C24×C4 |
| Frattini, cosocle | C24 | C25 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C26 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
13778 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 20 points:
(3,4);(1,2)(3,4);(10,12)(14,15)(17,18,19,20);(6,8)(10,12)(13,14,16,15);(5,6)(7,8)(9,10,11,12)(13,15,16,14)176 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 421204);