Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #19504 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
| direct product | C2×#19523 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#121 |
| commutator, Frattini, cosocle | C42 | C25 |
| prime radical | C24 | #121 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
1560 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);(5,11)(6,13)(8,15)(10,17)(12,14)(16,18)(19,20,24,28)(21,29,33,43)(22,26,34,40)(23,36,30,27)(25,37,39,32)(31,42,45,50)(35,47,44,41)(38,48,49,46);(3,4)(5,8)(6,10)(7,9)(11,15)(12,16)(13,17)(14,18)(19,21,30,37,24,33,23,32)(20,25,36,43,28,39,27,29)(22,31,44,48,34,45,35,46)(26,38,47,50,40,49,41,42);(3,5,7,12)(4,8,9,16)(6,14,13,11)(10,18,17,15)(19,22)(20,26)(21,31)(23,35)(24,34)(25,38)(27,41)(28,40)(29,42)(30,44)(32,46)(33,45)(36,47)(37,48)(39,49)(43,50);(4,9)(8,16)(10,17)(15,18)(22,34)(26,40)(31,45)(35,44)(38,49)(41,47)(42,50)(46,48)74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6261803);