Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #18992 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
| direct product | C2×#19013 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#120 |
| commutator, Frattini, cosocle | C42 | C25 |
| prime radical | C2×#3531 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
2630 conjugacy classes of subgroups.
| minimal degree | μ(G) = 50 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 50 points:
(1,2);(3,4)(5,12)(6,10)(7,15)(8,11)(9,13)(14,17)(16,18)(19,20,25,28)(21,31,34,43)(22,35,37,44)(23,30,24,29)(26,38,41,33)(27,39,42,36)(32,40,48,50)(45,46,49,47);(3,4)(5,9)(6,10)(7,15)(8,11)(12,16)(13,17)(14,18)(19,21,25,34)(20,26,28,41)(22,32,37,48)(23,38,24,33)(27,40,42,50)(29,43,30,31)(35,49,44,45)(36,47,39,46);(3,5)(4,9)(6,13)(7,16)(8,14)(10,17)(11,18)(12,15)(19,22)(20,27)(21,32)(23,36)(24,39)(25,37)(26,40)(28,42)(29,35)(30,44)(31,45)(33,46)(34,48)(38,47)(41,50)(43,49);(3,6,8,7)(4,10,11,15)(5,13,14,16)(9,17,18,12)(20,28)(26,41)(27,42)(29,30)(31,43)(35,44)(40,50)(45,49)68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6258593);