Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #40484 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
| direct product | C2×#41584 |
| 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 | |
4366 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);(5,9)(8,10)(11,12,16,13)(14,19,22,17)(15,18,23,20)(21,25,26,24)(27,28)(29,34)(30,38)(31,36)(32,33)(35,41)(37,40)(39,42);(3,4,7,6)(5,8,9,10)(12,13)(17,19)(18,20)(24,25)(28,33)(31,34)(35,38)(40,42);(3,5,7,9)(4,8,6,10)(11,13,16,12)(14,19,22,17)(15,20,23,18)(21,25,26,24)(27,29,32,36)(28,34,33,31)(30,39,41,37)(35,40,38,42);(3,6,7,4)(5,10,9,8)(11,14,16,22)(12,17,13,19)(15,21,23,26)(18,24,20,25)(27,30,32,41)(28,35,33,38)(29,37,36,39)(31,40,34,42);(11,15,16,23)(12,18,13,20)(14,21,22,26)(17,24,19,25)(27,31,32,34)(28,36,33,29)(30,40,41,42)(35,39,38,37)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7547300);