Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #10120 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
| direct product | C2×#10125 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #392 |
| commutator, Frattini, cosocle | C2×C42 | C24 |
| prime radical | C23×D4 | D4 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
1248 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)(5,11,12,18)(6,14,15,9)(8,16,17,13)(19,20)(21,30)(22,35)(23,40)(24,28)(25,29)(26,32)(27,38)(31,48)(33,43)(34,42)(36,47)(37,46)(39,45)(41,49)(44,50);(6,15)(9,14)(11,18)(13,16)(19,21)(20,26)(22,36)(23,33)(24,32)(25,34)(27,44)(28,42)(29,43)(30,40)(31,37)(35,41)(38,49)(39,50)(45,48)(46,47);(3,5)(4,8)(6,16)(7,12)(9,18)(10,17)(11,14)(13,15)(19,22,25,39)(20,27,29,46)(21,31,34,49)(23,37,24,38)(26,41,43,48)(28,45,40,35)(30,47,42,44)(32,36,33,50)62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 143799);