Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #10637 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
| direct product | C2×#10643 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #397 |
| 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 | |
730 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);(3,4,7,8,6,10,5,9)(11,12)(13,21,24,34)(14,18)(15,19)(16,20)(17,23,31,28)(22,37,25,36)(26,32)(27,33)(29,35)(30,39,41,38)(40,42);(11,13,16,24)(12,17,20,31)(14,25,27,22)(15,28,29,23)(18,30,33,41)(19,34,35,21)(26,38,40,39)(32,37,42,36);(4,8)(5,7)(9,10)(11,14,15,26,16,27,29,40)(12,18,19,32,20,33,35,42)(13,22,23,38,24,25,28,39)(17,30,21,36,31,41,34,37)74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 150212);