Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9755 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
| direct product | C2×#9755 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #390 |
| commutator | C42 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| prime radical | C24 | #127 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
1006 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);(4,7)(5,9)(8,10)(11,12)(13,17,22,31)(14,24)(15,27)(16,19)(18,25)(20,28)(21,33,34,42)(23,37,36,30)(26,38)(29,40,41,35)(32,39);(3,4,5,7,6,8,9,10)(11,13)(12,17,19,31)(14,21)(15,23)(16,22)(18,29,32,41)(20,30,27,37)(24,33,38,42)(25,35)(26,34)(28,36)(39,40);(11,14,15,25,16,26,28,39)(12,18,20,24,19,32,27,38)(13,21,23,35,22,34,36,40)(17,29,30,33,31,41,37,42)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 112728);