Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9279 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
| direct product | C2×#9291 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #386 |
| commutator | C42 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| prime radical | C2×#532 | C22 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
1364 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);(3,4,5,9,7,10,6,8)(11,12)(13,21)(14,25,27,18)(15,28)(16,20)(17,23)(19,30)(22,35,38,42)(24,36)(26,33,29,34)(31,37,41,39)(32,40);(4,8)(5,6)(9,10)(11,13,15,23,16,24,30,40)(12,17,19,21,20,32,28,36)(14,22,29,39,27,38,26,37)(18,31,34,35,25,41,33,42);(4,9)(5,6)(8,10)(11,14)(12,18)(13,22)(15,29)(16,27)(17,31)(19,34)(20,25)(21,35)(23,39)(24,38)(26,30)(28,33)(32,41)(36,42)(37,40)80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 106123);