Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3057 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C27×C4 |
| direct product | C2×#21659 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C22×D4 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| socle | C23 | C2×C4×D4 |
| prime radical | C2×#532 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C22×#97 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2062 conjugacy classes of subgroups.
| minimal degree | μ(G) = 42 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 42 points:
(1,2);(3,4)(5,8)(6,7)(9,10)(11,12,17,22,18,23,16,34)(13,24,28,36,29,40,27,19)(14,30,15,21,32,38,31,20)(25,37,33,42,41,35,26,39);(4,7)(8,10)(11,13,16,27,18,29,17,28)(12,19,22,24,23,36,34,40)(14,25,31,26,32,41,15,33)(20,35,30,39,21,37,38,42);(3,5,6,9)(4,8,7,10)(11,14,16,31,18,32,17,15)(12,20,22,30,23,21,34,38)(13,25,27,26,29,41,28,33)(19,35,24,39,36,37,40,42);(11,15,17,32,18,31,16,14)(12,21,22,38,23,20,34,30)(13,26,28,25,29,33,27,41)(19,35,24,39,36,37,40,42)128 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6280719);