Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #392 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C8 · Schur multiplier (H2) = C26 |
| direct product | C2×#13121 |
| subgroup | G/H | |
|---|---|---|
| center | C23×C8 | C23 |
| commutator | C23 | C23×C8 |
| Frattini, cosocle | C22×C8 | C24 |
| socle | C24 | C22×C8 |
| prime radical | C25 | C2×C8 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
966 conjugacy classes of subgroups.
| minimal degree | μ(G) = 26 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 26 points:
(1,2);(3,4)(5,6)(8,9)(11,12,15,18,16,19,23,25,14,13,17,20,22,21,24,26);(7,8,10,9)(11,13,15,20,16,21,23,26,14,12,17,18,22,19,24,25);(4,6)(7,9,10,8)(12,13)(18,20)(19,21)(25,26)176 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 402015);