Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #560 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C8 · Schur multiplier (H2) = C28 |
| direct product | C2×#33494 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C8 | C24 |
| commutator | C22 | C24×C8 |
| Frattini, cosocle | C2×C8 | C25 |
| socle | C23 | C23×C8 |
| prime radical | C2×#532 | C4 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C8 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
1578 conjugacy classes of subgroups.
| minimal degree | μ(G) = 42 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 42 points:
(1,2);(3,4)(5,9)(6,8)(7,10)(11,12,14,19,15,20,25,33,16,21,26,34,27,35,38,42)(13,17,23,28,24,29,37,39,22,18,30,31,36,32,40,41);(3,5)(4,7)(6,10)(8,9)(11,13)(12,17)(14,23)(15,24)(16,22)(18,21)(19,28)(20,29)(25,37)(26,30)(27,36)(31,34)(32,35)(33,39)(38,40)(41,42);(5,10)(7,9)(11,13)(12,18)(14,23)(15,24)(16,22)(17,21)(19,31)(20,32)(25,37)(26,30)(27,36)(28,34)(29,35)(33,41)(38,40)(39,42);(4,8)(7,9)(13,22)(17,18)(23,30)(24,36)(28,31)(29,32)(37,40)(39,41)152 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7529877);