Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #7855 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C24×C4 |
| direct product | C2×#8142 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C2×C42 |
| commutator | C2×C4 | C22×C42 |
| Frattini, cosocle | C2×C42 | C24 |
| socle | C23 | #376 |
| prime radical | C24 | #117 |
| LCS | G ▷ C2×C4 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
686 conjugacy classes of subgroups.
| minimal degree | μ(G) = 50 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 50 points:
(1,2);(3,4,7,10,6,9,14,17)(5,11,13,18,12,8,15,16)(19,20)(21,29,32,42)(22,26)(23,27)(24,28)(25,31,37,45)(30,41,44,49)(33,38)(34,39)(35,40)(36,43,47,50)(46,48);(3,5)(4,8)(6,12)(7,13)(9,11)(10,16)(14,15)(17,18)(19,21)(20,25,28,37)(22,30)(23,31)(24,32)(26,36,39,47)(27,29,40,42)(33,43)(34,44)(35,45)(38,41,48,49)(46,50);(5,12)(8,11)(13,15)(16,18)(19,22,23,33,24,34,35,46)(20,26,27,38,28,39,40,48)(21,30,31,43,32,44,45,50)(25,36,29,41,37,47,42,49)128 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 89025);