Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2668 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
| direct product | C2×#52305 |
| 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 | |
4918 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);(3,4,7,9)(5,10,6,8)(11,12)(13,16)(14,17)(15,18);(3,4)(5,10)(6,8)(7,9)(11,13,15,14)(12,16,18,17)(19,20)(21,24)(22,23)(25,26);(11,13,15,14)(12,17,18,16)(20,23)(24,26);(3,5,7,6)(4,8,9,10)(12,18)(16,17);(3,6,7,5)(4,8,9,10)(11,14,15,13)(12,17,18,16)(19,21,22,25)(20,24,23,26)128 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7540840);