Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #7221 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C22×C4 |
| direct product | C2×#7224 |
| 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 | |
412 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)(19,20,23,28,25,31,42,30)(21,32,34,45,37,47,36,50)(22,38,40,27,41,29,24,43)(26,35,44,33,46,49,48,39);(3,5)(4,8,10,15)(6,12)(7,13)(9,11,16,17)(14,18)(19,21,24,35,42,34,41,33,25,37,40,49,23,36,22,39)(20,26,29,45,28,48,43,32,31,46,38,50,30,44,27,47)80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 34471);