Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #10464 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
| direct product | C2×#10466 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #394 |
| commutator, Frattini, cosocle | C2×C42 | C24 |
| prime radical | C2×#532 | C22 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
1124 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)(5,7)(6,9)(8,10)(11,12)(13,22)(14,27)(15,32)(16,20)(17,21)(18,24)(19,30)(23,38)(25,34)(26,37)(28,33)(29,36)(31,41)(35,40)(39,42);(11,13,17,26)(12,18,21,34)(14,28,30,39)(15,25,16,24)(19,35,27,38)(20,37,32,22)(23,29,40,31)(33,41,42,36);(4,7)(5,10)(8,9)(11,14,15,29,17,30,16,31)(12,19,20,36,21,27,32,41)(13,23,24,39,26,40,25,28)(18,33,22,35,34,42,37,38)74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 147340);