Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #47826 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
| direct product | C2×#51493 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| prime radical | C25 | C24 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4886 conjugacy classes of subgroups.
| minimal degree | μ(G) = 34 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 34 points:
(1,2);(3,4)(5,9)(6,7)(8,10)(11,12,14,16)(13,15,17,18)(19,20,25,29)(21,30,23,26)(22,27,28,24)(31,32,33,34);(4,7)(8,9)(19,21,25,23)(20,26,29,30)(22,33,28,31)(24,34,27,32);(3,5)(4,8)(6,10)(7,9)(11,12)(13,15)(14,16)(17,18)(19,22,25,28)(20,27,29,24)(21,31,23,33)(26,34,30,32);(3,4)(5,8)(6,7)(9,10)(19,23,25,21)(20,26,29,30)(22,31,28,33)(24,34,27,32);(3,5)(4,8)(6,10)(7,9)(11,13,14,17)(12,15,16,18)(19,24,25,27)(20,28,29,22)(21,32,23,34)(26,33,30,31)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7614511);