Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #7262 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C24 |
| direct product | C2×#7265 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #340 |
| commutator | C2×C8 | C2×C42 |
| Frattini, cosocle | C23×C8 | C23 |
| prime radical | C25 | Q16 |
| LCS | G ▷ C2×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24×C4 ◁ C24×C8 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
830 conjugacy classes of subgroups.
| minimal degree | μ(G) = 42 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 42 points:
(1,2);(3,4)(5,7)(6,9)(8,10)(11,12,15,20,17,22,16,21)(13,23,25,40,27,41,26,37)(14,28,30,39,32,38,31,19)(18,33,34,24,36,29,35,42);(4,7,9,8)(11,13,15,26,17,27,16,25)(12,18,20,35,22,36,21,34)(14,29,30,24,32,33,31,42)(19,37,28,41,39,40,38,23)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 34723);