Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #19186 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
| direct product | C2×#19188 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#121 |
| commutator, Frattini, cosocle | C42 | C25 |
| prime radical | C2×#532 | C22 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
2110 conjugacy classes of subgroups.
| minimal degree | μ(G) = 42 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 42 points:
(1,2);(3,4)(5,7)(6,9)(8,10)(11,12,16,20)(13,21,25,35)(14,18,26,32)(15,28,22,19)(17,29,31,24)(23,34,37,42)(27,39,36,33)(30,40,41,38);(11,13,22,29,16,25,15,24)(12,17,28,35,20,31,19,21)(14,23,36,40,26,37,27,38)(18,30,39,42,32,41,33,34);(4,7)(5,10)(8,9)(11,14)(12,18)(13,23)(15,27)(16,26)(17,30)(19,33)(20,32)(21,34)(22,36)(24,38)(25,37)(28,39)(29,40)(31,41)(35,42);(14,26)(18,32)(23,37)(27,36)(30,41)(33,39)(34,42)(38,40)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6260865);