Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9755 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
| direct product | C2×#9772 |
| 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 | |
1302 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);(4,7)(5,8)(9,10)(11,12,16,21,17,20,15,29)(13,18,24,33,23,34,30,32)(14,25,27,35,28,39,26,19)(22,36,38,31,37,41,40,42);(3,4,8,10,6,9,5,7)(11,13,15,24,17,23,16,30)(12,18,21,32,20,34,29,33)(14,22,26,38,28,37,27,40)(19,31,25,36,35,42,39,41);(11,14,16,27,17,28,15,26)(12,19,29,39,20,35,21,25)(13,22,30,40,23,37,24,38)(18,31,33,41,34,42,32,36)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 112839);