Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2215 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
| direct product | C2×#13204 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C2×D8 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C22×C8 | C24 |
| prime radical | C23×D8 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C22×C42 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
1410 conjugacy classes of subgroups.
| minimal degree | μ(G) = 42 |
| transitive degree | μt(G) = 256 |
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)(16,29)(17,23)(19,22)(20,27)(24,33)(25,40)(28,41)(30,39)(31,42)(32,37)(34,36)(35,38);(8,10)(11,13,22,36,16,25,28,39,15,24,37,42,27,38,14,23)(12,17,26,35,20,31,32,33,18,30,41,40,29,34,19,21);(3,4,5,6)(12,18)(17,30)(19,32)(20,29)(21,33)(26,41)(31,34)(35,40)152 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 402311);