Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #10370 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
| direct product | C2×#10630 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | #124 |
| commutator, Frattini, cosocle | C2×C42 | C24 |
| socle | C23 | #397 |
| prime radical | C24 | #123 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
646 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)(5,8)(6,9)(7,10)(11,18)(12,16)(13,15)(14,17)(19,20)(21,29,32,42)(22,26)(23,27)(24,28)(25,31,39,36)(30,45,33,44)(34,40)(35,41)(37,43)(38,47,49,46)(48,50);(3,5,6,12)(4,8,9,16)(7,11,14,13)(10,18,17,15)(19,21)(20,25)(22,33)(23,36)(24,32)(26,38)(27,42)(28,39)(29,43)(30,35)(31,37)(34,46)(40,45)(41,49)(44,50)(47,48);(4,9)(5,11)(7,14)(8,15)(12,13)(16,18)(19,22,23,34,24,35,37,48)(20,26,27,40,28,41,43,50)(21,30,31,46,32,33,36,47)(25,38,29,44,39,49,42,45)80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 150110);