Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #7217 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C22×C4 |
| direct product | C2×#7220 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #340 |
| commutator | C2×C8 | C2×C42 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| prime radical | C24 | #103 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
422 conjugacy classes of subgroups.
| minimal degree | μ(G) = 50 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 3 generators on 50 points:
(1,2);(3,4)(5,11,13,17)(6,9)(7,10)(8,12,15,18)(14,16)(19,20,23,29,26,31,25,41)(21,32,34,47,36,48,39,43)(22,30,24,40,38,46,37,28)(27,35,42,33,45,49,44,50);(3,5)(4,8,10,15)(6,12)(7,13)(9,11,16,17)(14,18)(19,21,24,35,23,39,38,50,26,36,37,49,25,34,22,33)(20,27,30,43,41,42,28,32,31,45,46,47,29,44,40,48)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 34455);