Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #52308 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
| direct product | C2×#52787 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| prime radical | C25 | C24 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4116 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);(3,4,8,6)(5,11,12,9)(7,14,15,10)(13,16,18,17)(22,30)(25,32)(28,34)(31,33)(36,40)(41,43)(42,47)(48,49);(4,6)(5,12)(10,14)(13,18)(19,20,23,26)(21,24,29,27)(22,25,30,32)(28,31,34,33)(35,36,39,40)(37,43,45,41)(38,42,46,47)(44,49,50,48);(3,5,8,12)(4,9,6,11)(7,13,15,18)(10,16,14,17)(19,20)(21,27)(22,25)(23,26)(24,29)(28,33)(30,32)(31,34)(35,37,39,45)(36,41,40,43)(38,44,46,50)(42,48,47,49);(3,6,8,4)(5,9,12,11)(7,14,15,10)(13,16,18,17)(19,21)(20,24)(22,28)(23,29)(25,31)(26,27)(30,34)(32,33)(35,36)(37,41)(38,42)(39,40)(43,45)(44,48)(46,47)(49,50);(3,7,8,15)(4,10,6,14)(5,13,12,18)(9,16,11,17)(19,22)(20,25)(21,28)(23,30)(24,31)(26,32)(27,33)(29,34)(35,38,39,46)(36,42,40,47)(37,44,45,50)(41,48,43,49)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7999738);