Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #37436 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
| direct product | C2×#37577 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4610 conjugacy classes of subgroups.
| minimal degree | μ(G) = 42 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 42 points:
(1,2);(5,9)(8,10)(12,16)(14,22)(17,19)(20,26)(27,28,32,33)(29,31,36,34)(30,38,40,35)(37,41,42,39);(3,4,7,6)(5,8,9,10)(12,16)(17,19)(18,23)(24,25)(28,33)(29,36)(35,38)(37,42);(3,5)(4,8)(6,10)(7,9)(11,12)(13,19)(14,18)(15,16)(17,21)(20,25)(22,23)(24,26)(27,29,32,36)(28,34,33,31)(30,37,40,42)(35,41,38,39);(3,6,7,4)(5,10,9,8)(11,13,15,21)(12,17,16,19)(14,20,22,26)(18,24,23,25)(27,30,32,40)(28,35,33,38)(29,37,36,42)(31,39,34,41);(11,14)(12,18)(13,20)(15,22)(16,23)(17,24)(19,25)(21,26)(27,31,32,34)(28,36,33,29)(30,39,40,41)(35,42,38,37)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7541020);