Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #10631 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
| direct product | C2×#10635 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #397 |
| commutator, Frattini, cosocle | C2×C42 | C24 |
| prime radical | C24 | #124 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
720 conjugacy classes of subgroups.
| minimal degree | μ(G) = 42 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 42 points:
(1,2);(4,7)(8,10)(11,12,16,21,17,22,15,20)(13,23,26,18,27,34,25,35)(14,28,30,36,31,37,32,19)(24,39,40,42,29,33,41,38);(4,8)(6,9)(7,10)(11,13,17,27)(12,18,22,35)(14,29,31,24)(15,26,16,25)(19,33,36,39)(20,34,21,23)(28,42,37,38)(30,40,32,41);(3,4)(5,7)(6,10)(8,9)(11,14,15,30,17,31,16,32)(12,19,20,28,22,36,21,37)(13,24,25,40,27,29,26,41)(18,33,23,38,35,39,34,42)68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 150150);