Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2218 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
| direct product | C2×#13207 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C24 | C2×D8 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C22×C8 | C24 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C22×C42 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
1326 conjugacy classes of subgroups.
| minimal degree | μ(G) = 42 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 4 generators on 42 points:
(1,2);(4,6)(7,8,9,10)(11,12)(13,21)(14,26)(15,18)(16,30)(17,24)(19,29)(20,27)(22,34)(23,36)(25,42)(28,31)(32,39)(33,38)(35,41)(37,40);(8,10)(11,13,22,38,16,25,14,24,15,23,29,41,27,40,28,39)(12,17,31,42,20,33,19,21,18,32,26,37,30,35,34,36);(3,4,5,6)(12,18)(13,23)(19,34)(20,30)(24,39)(25,40)(26,31)(38,41)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 402315);