Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #45438 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C213 |
| direct product | C2×#46238 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| prime radical | C22×#562 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
6806 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);(4,6)(8,12)(9,15)(14,19)(16,21)(23,24)(25,31)(26,34)(27,30)(28,33)(29,35)(32,36)(37,38);(8,12)(9,15)(10,17)(20,22)(23,24,27,30)(25,28,33,31)(26,34,35,29)(32,37,38,36);(3,4,5,6)(7,8)(9,13)(10,14)(11,12)(15,18)(16,20)(17,19)(21,22)(23,24)(25,28)(26,29)(27,30)(31,33)(32,36)(34,35)(37,38);(7,9,11,15)(8,13,12,18)(10,16,17,21)(14,20,19,22)(23,25)(24,28)(26,32)(27,33)(29,36)(30,31)(34,37)(35,38);(7,10)(8,14)(9,16)(11,17)(12,19)(13,20)(15,21)(18,22)(23,26,27,35)(24,29,30,34)(25,32,33,38)(28,36,31,37)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7610118);