Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2184 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
| direct product | C2×#13015 |
| 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 | |
1358 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)(11,12,15,21,31,40,29,18,17,23,14,20,16,22,32,36)(13,24,26,42,34,39,30,38,28,35,25,37,27,41,33,19);(3,4,5,6)(12,18)(14,29)(15,32)(16,31)(19,35)(20,22)(21,40)(23,36)(24,38)(25,30)(26,33)(27,34)(37,41)(39,42);(8,10)(11,13,17,28)(12,19,23,38)(14,30,15,33)(16,34,31,27)(18,35,36,24)(20,39,21,41)(22,42,40,37)(25,32,26,29)116 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 401692);