Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of D32 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C22×C4 |
| direct product | C4×#1424 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | D16 |
| commutator | C16 | C2×C42 |
| Frattini, cosocle | C22×C16 | C23 |
| socle, prime radical | C23 | C2×D16 |
| LCS | G ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C2×C42 ◁ C2×C4×C8 ◁ C2×C4×C16 ◁ G | |
| LECS | G ▷ C22×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
226 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 3 generators on 40 points:
(1,2,3,4);(5,6,7,8)(10,15)(11,20)(12,21)(13,24)(16,23)(17,32)(18,35)(19,29)(22,31)(25,38)(26,27)(28,33)(30,39)(34,40)(36,37);(9,10,16,30,12,17,22,28,24,33,37,32,38,39,11,15,14,19,20,27,25,34,36,35,13,18,31,40,21,26,23,29)152 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 59187);