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×#62383 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C2×C4 | C26 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#533 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
5946 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)(21,29)(22,33)(23,27)(24,38)(25,31)(26,36)(28,37)(30,44)(32,40)(34,48)(35,47)(39,49)(41,45)(42,50)(43,46);(3,5,7,13)(4,8,10,11)(6,14,15,12)(9,17,18,16)(19,21,23,31)(20,25,27,29)(22,34,35,30)(24,32,37,46)(26,41,42,39)(28,40,38,43)(33,44,47,48)(36,49,50,45);(19,22,24,36,23,35,37,50)(20,26,28,33,27,42,38,47)(21,30,32,45,31,34,46,49)(25,39,40,48,29,41,43,44);(3,6,7,15)(4,9,10,18)(5,12,13,14)(8,16,11,17)(21,31)(25,29)(30,34)(32,46)(39,41)(40,43)(44,48)(45,49);(4,10)(8,11)(9,18)(16,17)(20,27)(21,31)(26,42)(28,38)(30,34)(32,46)(33,47)(45,49)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10485620);