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×#50189 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| prime radical | C26 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4874 conjugacy classes of subgroups.
| minimal degree | μ(G) = 42 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 42 points:
(1,2);(5,9)(7,10)(11,12,16,14)(13,19,21,17)(15,22,23,18)(20,24,26,25)(27,28,32,35)(29,31,34,40)(30,33,37,41)(36,39,38,42);(3,4,6,8)(5,7,9,10)(12,14)(17,19)(18,22)(24,25)(29,34)(30,37)(31,40)(33,41);(3,5)(4,7)(6,9)(8,10)(11,13,16,21)(12,17,14,19)(15,20,23,26)(18,24,22,25)(27,29,32,34)(28,31,35,40)(30,38,37,36)(33,42,41,39);(11,14,16,12)(13,19,21,17)(15,22,23,18)(20,25,26,24)(27,30)(28,33)(29,36)(31,39)(32,37)(34,38)(35,41)(40,42);(11,15)(12,18)(13,20)(14,22)(16,23)(17,24)(19,25)(21,26)(27,31)(28,34)(29,35)(30,39)(32,40)(33,38)(36,41)(37,42)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7613094);