Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #19186 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
| direct product | C2×#19220 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#121 |
| commutator, Frattini, cosocle | C42 | C25 |
| prime radical | C2×#121 | D4 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
1590 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)(19,20,24,28)(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);(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)(4,9)(6,13)(7,16)(8,14)(10,17)(11,18)(12,15)(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);(3,6,8,7)(4,10,11,15)(5,13,14,16)(9,17,18,12)(22,34)(26,40)(31,45)(35,44)(38,49)(41,47)(42,50)(46,48)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6260903);