Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #10128 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
| direct product | C2×#10134 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #392 |
| commutator, Frattini, cosocle | C2×C42 | C24 |
| prime radical | C2×#532 | C22 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
1252 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);(5,11)(6,13)(8,16)(9,14)(12,15)(17,18)(19,20,24,28)(21,29,32,33)(22,27,23,26)(25,31,34,30);(3,4,7,10)(5,8,12,17)(6,14,13,9)(11,18,15,16)(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)(11,13)(14,16)(20,26)(21,30)(22,23)(27,28)(29,33)(31,32)62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 143914);