Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #53390 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
| direct product | C2×#56946 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| prime radical | C2×#392 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4392 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);(4,8)(5,11)(10,15)(12,17)(19,20)(21,25)(22,30)(23,24)(26,31)(27,28)(29,34)(32,33)(35,36,39,40)(37,43,44,41)(38,46,47,42)(45,48,50,49);(4,8)(9,14)(10,15)(16,18)(20,24)(21,27)(22,31)(32,34)(36,40)(37,44)(42,46)(45,50);(3,4)(5,9)(6,10)(7,8)(11,14)(12,16)(13,15)(17,18)(19,20)(21,28)(22,26)(23,24)(25,27)(29,34)(30,31)(32,33)(35,37,39,44)(36,41,40,43)(38,45,47,50)(42,48,46,49);(3,5)(4,9)(6,12)(7,11)(8,14)(10,16)(13,17)(15,18)(19,21)(20,25)(22,29)(23,27)(24,28)(26,32)(30,34)(31,33)(35,36)(37,41)(38,42)(39,40)(43,44)(45,48)(46,47)(49,50);(3,6,7,13)(4,10,8,15)(5,12,11,17)(9,16,14,18)(19,22,23,31)(20,26,24,30)(21,29,27,33)(25,32,28,34)(35,38)(36,42)(37,45)(39,47)(40,46)(41,48)(43,49)(44,50)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8006733);