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×#62376 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C2×C4 | C26 |
| prime radical | C2×#3259 | C2 |
| 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)(24,37)(25,31)(26,35)(28,36)(30,43)(32,39)(34,48)(38,50)(40,49)(41,45)(42,46)(44,47);(3,5,7,13)(4,8,10,11)(6,14,15,12)(9,17,18,16)(19,21,23,31)(20,25,27,29)(22,30,34,44)(24,32,36,46)(26,38,40,41)(28,39,37,42)(33,47,48,43)(35,45,49,50);(19,22,24,35,23,34,36,49)(20,26,28,33,27,40,37,48)(21,30,32,45,31,44,46,50)(25,38,39,47,29,41,42,43);(3,6,7,15)(4,9,10,18)(5,12,13,14)(8,16,11,17)(21,31)(25,29)(30,44)(32,46)(38,41)(39,42)(43,47)(45,50);(4,10)(8,11)(9,18)(16,17)(20,27)(21,31)(26,40)(28,37)(30,44)(32,46)(33,48)(45,50)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10485612);