Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #1347 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C22×C4 |
| direct product | C2×#7289 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | #102 |
| commutator | C2×C8 | C2×C42 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C23 | #340 |
| prime radical | C24 | #106 |
| LCS | G ▷ C2×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ 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);(3,4,6,9,7,10,14,17)(5,11,13,18,12,8,15,16)(19,20)(21,30)(22,35)(23,38)(24,40)(25,29)(26,31)(27,37)(28,36)(32,42)(33,50)(34,46)(39,43)(41,49)(44,48)(45,47);(3,5)(4,8,10,11)(6,13)(7,12)(9,16,17,18)(14,15)(19,21,23,32,36,47,37,48,25,34,22,31,24,33,39,49)(20,26,27,30,40,44,38,50,29,42,43,46,28,41,35,45)116 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 35071);