Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #509 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C24×C4 |
| direct product | C4×#2161 |
| subgroup | G/H | |
|---|---|---|
| center | C42 | C2×D8 |
| commutator | C8 | C22×C42 |
| Frattini, cosocle | C22×C8 | C24 |
| socle | C22 | C22×#107 |
| prime radical | C24 | C2×Q16 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C42 ◁ C22×C42 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
732 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2,3,4);(5,6,9,14,11,15,10,25)(7,16,20,32,18,30,21,35)(8,22,23,29,24,36,17,13)(12,26,28,34,27,33,31,19);(5,7,17,33,10,21,24,34,11,18,23,26,9,20,8,19)(6,12,22,32,14,28,29,30,15,27,36,35,25,31,13,16);(7,18)(12,27)(16,30)(19,34)(20,21)(26,33)(28,31)(32,35)128 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 400347);