Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #10012 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
| direct product | C2×#10035 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | #127 |
| commutator, Frattini, cosocle | C2×C42 | C24 |
| socle | C23 | #392 |
| prime radical | C2×#532 | C22 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
1358 conjugacy classes of subgroups.
| minimal degree | μ(G) = 34 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 34 points:
(1,2);(3,4)(5,11)(6,14)(7,10)(8,13)(9,16)(12,18)(15,17)(19,20,24,28)(21,29,32,33)(22,27,23,26)(25,31,34,30);(8,17)(11,18)(19,21,24,32)(20,25,28,34)(22,31,23,30)(26,33,27,29);(3,5)(4,8)(6,15)(7,12)(9,18)(10,17)(11,14)(13,16)(21,30)(22,23)(25,29)(26,27)(31,32)(33,34)80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 139941);