Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2184 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
| direct product | C2×#11650 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C2×D8 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C4×C8 | C24 |
| socle | C23 | C2×#102 |
| prime radical | C24 | #102 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C22×C42 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
774 conjugacy classes of subgroups.
| minimal degree | μ(G) = 50 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 50 points:
(1,2);(3,4,6,9,7,10,13,16)(5,8,12,15,11,14,17,18)(19,20)(21,30)(22,27)(23,36)(24,40)(25,29)(26,32)(28,38)(31,44)(33,50)(34,47)(35,45)(37,43)(39,46)(41,48)(42,49);(3,5)(4,8)(6,12)(7,11)(9,15)(10,14)(13,17)(16,18)(19,21,25,34)(20,26,29,42)(22,33,24,31)(23,37,39,48)(27,43,40,41)(28,44,45,50)(30,46,47,36)(32,35,49,38);(5,11)(8,14)(12,17)(15,18)(19,22,25,24)(20,27,29,40)(21,31,34,33)(23,35,39,38)(26,41,42,43)(28,36,45,46)(30,44,47,50)(32,48,49,37)116 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 388492);