Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9758 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
| direct product | C2×#9779 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #390 |
| commutator | C42 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| prime radical | C24 | #124 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
1238 conjugacy classes of subgroups.
| minimal degree | μ(G) = 42 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 42 points:
(1,2);(3,4)(5,9)(6,10)(7,8)(11,12,16,21,17,20,15,30)(13,18,24,34,23,35,31,33)(14,25,28,37,29,41,27,19)(22,38,39,32,26,40,42,36);(3,5,9,4,7,10,6,8)(11,13,15,24,17,23,16,31)(12,18,21,33,20,35,30,34)(14,26,27,42,29,22,28,39)(19,36,25,40,37,32,41,38);(4,8)(5,10)(11,14,15,27,17,29,16,28)(12,19,21,25,20,37,30,41)(13,22,24,39,23,26,31,42)(18,32,33,38,35,36,34,40)80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 112852);