Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #47826 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C213 |
| direct product | C2×#51544 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| prime radical | C25 | C24 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
6550 conjugacy classes of subgroups.
| minimal degree | μ(G) = 38 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 38 points:
(1,2);(3,4,5,6)(24,28)(26,34)(29,31)(32,38);(8,12)(9,15)(10,18)(20,22)(23,24,27,28)(25,29,33,31)(26,35,34,30)(32,37,38,36);(3,4)(5,6)(7,8)(9,16)(10,14)(11,12)(13,15)(17,22)(18,19)(20,21)(23,24)(25,31)(26,30)(27,28)(29,33)(32,37)(34,35)(36,38);(7,9)(8,13)(10,17)(11,15)(12,16)(14,20)(18,21)(19,22)(23,25,27,33)(24,29,28,31)(26,32,34,38)(30,36,35,37);(7,10,11,18)(8,14,12,19)(9,17,15,21)(13,20,16,22)(23,26,27,34)(24,30,28,35)(25,32,33,38)(29,36,31,37)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7614818);