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×#52861 |
| 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 | |
4186 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)(6,10)(7,8)(9,13)(12,17)(14,15)(16,18)(19,20,24,25)(21,26,28,23)(22,30,31,27)(29,32,34,33)(38,46)(41,48)(44,50)(47,49);(4,8)(9,11)(10,15)(16,17)(20,25)(21,28)(22,31)(32,33)(35,36,39,42)(37,40,45,43)(38,41,46,48)(44,47,50,49);(3,5,7,13)(4,9,8,11)(6,12,14,18)(10,16,15,17)(19,21)(20,26)(22,29)(23,25)(24,28)(27,33)(30,32)(31,34)(35,36)(37,43)(38,41)(39,42)(40,45)(44,49)(46,48)(47,50);(3,4)(5,9)(6,10)(7,8)(11,13)(12,16)(14,15)(17,18)(19,22,24,31)(20,27,25,30)(21,29,28,34)(23,32,26,33)(35,37)(36,40)(38,44)(39,45)(41,47)(42,43)(46,50)(48,49);(3,6,7,14)(4,10,8,15)(5,12,13,18)(9,16,11,17)(19,23,24,26)(20,21,25,28)(22,32,31,33)(27,29,30,34)(35,38)(36,41)(37,44)(39,46)(40,47)(42,48)(43,49)(45,50)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7999837);