Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #392 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C26 |
| direct product | C2×#8135 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C42 | C23 |
| commutator | C23 | C22×C42 |
| Frattini, cosocle | C2×C42 | C24 |
| socle, prime radical | C24 | C2×C42 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
914 conjugacy classes of subgroups.
| minimal degree | μ(G) = 42 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 4 generators on 42 points:
(1,2);(3,4)(5,8)(6,7)(9,10)(11,12,14,18,15,17,21,24)(13,16,20,22,19,23,25,26)(27,28)(29,32)(30,34)(31,33)(35,39)(36,38)(37,40)(41,42);(4,7)(8,10)(11,13)(12,16)(14,20)(15,19)(17,23)(18,22)(21,25)(24,26)(27,29,30,35,31,36,37,41)(28,32,34,39,33,38,40,42);(3,5,6,9)(4,8,7,10)(12,17)(13,19)(18,24)(20,25)(28,33)(32,38)(34,40)(39,42)176 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 89004);