Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #534 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C8 · Schur multiplier (H2) = C28 |
| direct product | C2×#33495 |
| 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 | |
1600 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)(11,12,15,20,16,21,27,35,14,19,25,33,26,34,38,42)(13,18,23,31,24,32,37,41,22,17,28,29,36,30,39,40);(3,5)(4,7)(6,10)(8,9)(11,13)(12,17)(14,22)(15,23)(16,24)(18,19)(20,29)(21,30)(25,28)(26,36)(27,37)(31,33)(32,34)(35,40)(38,39)(41,42);(5,10)(7,9)(11,13)(12,18)(14,22)(15,23)(16,24)(17,19)(20,31)(21,32)(25,28)(26,36)(27,37)(29,33)(30,34)(35,41)(38,39)(40,42);(4,8)(7,9)(12,19)(13,22)(20,33)(21,34)(23,28)(24,36)(35,42)(37,39)176 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7529878);