Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #62265 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C213 |
| direct product | C2×#62374 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C2×C4 | C26 |
| prime radical | C24 | C22×D4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#533 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
5838 conjugacy classes of subgroups.
| minimal degree | μ(G) = 50 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 50 points:
(1,2);(3,4)(5,11)(6,9)(7,10)(8,13)(12,17)(14,16)(15,18)(19,20,24,28)(21,25,30,38)(22,32,35,47)(23,36,33,27)(26,40,41,34)(29,42,44,50)(31,46,43,39)(37,48,49,45);(3,5,7,13)(4,8,10,11)(6,14,15,12)(9,17,18,16)(19,21)(20,25)(22,29)(23,31)(24,30)(26,37)(27,39)(28,38)(32,42)(33,43)(34,45)(35,44)(36,46)(40,48)(41,49)(47,50);(19,22,33,40,24,35,23,34)(20,26,36,47,28,41,27,32)(21,29,43,48,30,44,31,45)(25,37,46,50,38,49,39,42);(3,6,7,15)(4,9,10,18)(5,12,13,14)(8,16,11,17);(4,10)(8,11)(9,18)(16,17)(21,30)(25,38)(29,44)(31,43)(37,49)(39,46)(42,50)(45,48)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10485609);