Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #560 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C4 · Schur multiplier (H2) = C213 |
| direct product | C2×C4×#564 |
| subgroup | G/H | |
|---|---|---|
| center | C23×C4 | C24 |
| commutator | C22 | C25×C4 |
| Frattini, cosocle | C23 | C26 |
| socle, prime radical | C24 | C25 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
6406 conjugacy classes of subgroups.
| minimal degree | μ(G) = 38 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 6 generators on 38 points:
(1,2);(3,4,5,6);(9,15)(10,18)(12,16)(13,20)(23,24,27,30)(25,28,33,31)(26,34,35,29)(32,37,38,36);(7,8,11,14)(9,16,15,12)(10,13,18,20)(17,22,21,19)(23,24)(25,31)(26,34)(27,30)(28,33)(29,35)(32,36)(37,38);(7,9,11,15)(8,12,14,16)(10,17,18,21)(13,19,20,22)(23,25)(24,28)(26,32)(27,33)(29,36)(30,31)(34,37)(35,38);(7,10)(8,13)(9,17)(11,18)(12,19)(14,20)(15,21)(16,22)(23,26,27,35)(24,29,30,34)(25,32,33,38)(28,36,31,37)152 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7533133);