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×#57823 |
| 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 | |
4148 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);(3,4,7,10)(5,11,12,8)(6,9,14,16)(13,17,18,15)(19,20)(21,27)(22,30)(23,29)(24,25)(26,31)(28,33)(32,34)(35,36)(37,44)(38,41)(39,47)(40,43)(42,48)(45,49)(46,50);(5,12)(8,11)(13,18)(15,17)(20,25)(22,31)(23,27)(28,32)(35,37,40,38)(36,41,43,44)(39,45,48,46)(42,49,47,50);(3,5)(4,8)(6,13)(7,12)(9,15)(10,11)(14,18)(16,17)(19,21)(20,23)(22,32)(24,29)(25,27)(26,34)(28,31)(30,33)(35,38,40,37)(36,41,43,44)(39,46,48,45)(42,49,47,50);(3,4)(5,8)(6,9)(7,10)(11,12)(13,15)(14,16)(17,18)(19,22)(20,26)(21,28)(23,33)(24,31)(25,30)(27,34)(29,32)(35,36)(37,44)(38,41)(39,42)(40,43)(45,50)(46,49)(47,48);(3,6,7,14)(4,9,10,16)(5,13,12,18)(8,15,11,17)(19,23)(20,21)(22,33)(24,27)(25,29)(26,28)(30,32)(31,34)(35,39,40,48)(36,42,43,47)(37,45,38,46)(41,49,44,50)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8008627);