Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #10440 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
| direct product | C2×#10447 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #394 |
| commutator, Frattini, cosocle | C2×C42 | C24 |
| prime radical | C24 | #123 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
1094 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)(5,7)(6,8)(9,10)(11,12,16,20)(13,21,24,36)(14,25,28,18)(15,29,31,19)(17,30,33,23)(22,38,26,37)(27,35,42,40)(32,39,34,41);(4,7)(8,9)(11,13,16,24)(12,17,20,33)(14,26,28,22)(15,30,31,23)(18,34,25,32)(19,36,29,21)(27,39,42,41)(35,37,40,38);(4,8)(6,10)(7,9)(11,14,15,27,16,28,31,42)(12,18,19,35,20,25,29,40)(13,22,23,39,24,26,30,41)(17,32,21,37,33,34,36,38)74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 147104);