Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32712 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
| direct product | C2×#32756 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C22×#121 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C22×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1788 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);(3,4)(5,7)(6,8)(9,10)(11,12,14,17)(13,16,18,15)(19,20,23,25)(21,24,29,32)(22,26,30,34)(27,31,28,33);(3,5)(4,7)(6,9)(8,10)(11,13,14,18)(12,15,17,16)(19,21,23,29)(20,24,25,32)(22,27,30,28)(26,31,34,33);(4,8)(7,10)(11,13)(12,16)(14,18)(15,17)(21,27)(22,30)(24,31)(26,34)(28,29)(32,33);(5,9)(7,10)(13,18)(15,16)(20,25)(24,32)(26,34)(31,33)68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6538510);