Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2403 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
| direct product | C2×#34951 |
| subgroup | G/H | |
|---|---|---|
| center | C23×C4 | C24 |
| commutator, Frattini, cosocle | C23 | C26 |
| socle, prime radical | C24 | C25 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
5678 conjugacy classes of subgroups.
| minimal degree | μ(G) = 26 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 6 generators on 26 points:
(1,2);(3,4)(5,8)(6,7)(9,10)(11,12,15,17)(13,16,14,18);(4,7)(8,10)(11,13,15,14)(12,16,17,18)(19,20,23,26)(21,25,22,24);(11,14,15,13)(12,16,17,18)(19,21,23,22)(20,24,26,25);(19,22,23,21)(20,25,26,24);(3,5,6,9)(4,8,7,10)152 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7535317);