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×#51302 |
| 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)(14,17,22,19)(15,20,23,18)(21,25,26,24)(27,28,32,35)(29,36,37,33)(30,34,40,31)(38,42,41,39);(3,4,7,6)(5,8,9,10)(12,13)(15,23)(17,19)(21,26)(29,37)(30,40)(31,34)(33,36);(3,5)(4,8)(6,10)(7,9)(11,13,16,12)(14,17,22,19)(15,20,23,18)(21,24,26,25)(27,29)(28,33)(30,41)(31,39)(32,37)(34,42)(35,36)(38,40);(3,4,7,6)(5,8,9,10)(11,14,16,22)(12,17,13,19)(15,21,23,26)(18,24,20,25)(27,30)(28,34)(29,38)(31,35)(32,40)(33,39)(36,42)(37,41);(3,6,7,4)(5,10,9,8)(11,15)(12,18)(13,20)(14,21)(16,23)(17,24)(19,25)(22,26)(27,31,32,34)(28,30,35,40)(29,39,37,42)(33,41,36,38)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7614434);