Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #47826 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
| direct product | C2×#50517 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4398 conjugacy classes of subgroups.
| minimal degree | μ(G) = 42 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 6 generators on 42 points:
(1,2);(4,8)(9,10)(11,12,16,20)(13,15,19,17)(14,18,22,26)(21,24,23,25)(27,28)(29,36)(30,38)(31,32)(33,35)(34,41)(37,40)(39,42);(13,19)(14,22)(15,17)(18,26)(27,29,33,31)(28,32,35,36)(30,39,41,37)(34,42,38,40);(3,4)(5,9)(6,10)(7,8)(11,13,16,19)(12,17,20,15)(14,23,22,21)(18,24,26,25)(27,30,33,41)(28,34,35,38)(29,37,31,39)(32,40,36,42);(3,5,7,6)(4,9,8,10)(11,14,16,22)(12,18,20,26)(13,21,19,23)(15,24,17,25)(27,31,33,29)(28,32,35,36)(30,37,41,39)(34,42,38,40);(3,6,7,5)(4,10,8,9)(11,15,16,17)(12,19,20,13)(14,24,22,25)(18,23,26,21)(27,32)(28,29)(30,40)(31,35)(33,36)(34,37)(38,39)(41,42)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7613219);