Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #16249 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
| direct product | C2×#16706 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C25 | C24 |
| commutator, Frattini, cosocle | C24 | C25 |
| prime radical | C26 | C23 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C25 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
4126 conjugacy classes of subgroups.
| minimal degree | μ(G) = 34 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 34 points:
(1,2);(4,7)(5,9)(11,12,14,16)(13,17,18,15)(19,20,23,24)(21,22,25,26)(28,29)(32,33);(13,18)(15,17)(20,24)(21,25)(27,28,31,29)(30,32,34,33);(3,4,6,7)(5,8,9,10)(11,13,14,18)(12,15,16,17)(19,21)(20,22)(23,25)(24,26)(27,29,31,28)(30,32,34,33);(3,5)(4,8)(6,9)(7,10)(11,12)(13,15)(14,16)(17,18)(19,22)(20,25)(21,24)(23,26)(27,30)(28,32)(29,33)(31,34)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 518633);