Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #62862 |
| automorphism | ? |
| homology | Abelianization (H1) = C27 · Schur multiplier (H2) = C219 |
| direct product | C2×#62889 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C26 |
| commutator, Frattini, cosocle | C22 | C27 |
| prime radical | C2×#3455 | C2 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C22 ▷ C1 | |
30734 conjugacy classes of subgroups.
| minimal degree | μ(G) = 50 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 7 generators on 50 points:
(1,2);(3,4,8,10)(5,11,13,6)(7,15,16,9)(12,14,18,17)(19,20,24,25)(21,29,32,26)(22,27,33,37)(23,35,36,28)(30,41,43,38)(31,39,45,42)(34,47,46,40)(44,48,50,49);(3,5,8,13)(4,6,10,11)(7,12,16,18)(9,14,15,17)(20,25)(22,33)(26,29)(28,35)(30,43)(34,46)(39,42)(44,50);(3,6)(4,5)(7,14)(8,11)(9,12)(10,13)(15,18)(16,17)(19,20)(21,29)(22,27)(23,28)(24,25)(26,32)(30,41)(31,42)(33,37)(34,40)(35,36)(38,43)(39,45)(44,49)(46,47)(48,50);(3,7,8,16)(4,9,10,15)(5,12,13,18)(6,14,11,17)(19,21,24,32)(20,26,25,29)(22,30,33,43)(23,31,36,45)(27,38,37,41)(28,39,35,42)(34,44,46,50)(40,48,47,49);(19,22,24,33)(20,27,25,37)(21,30,32,43)(23,34,36,46)(26,38,29,41)(28,40,35,47)(31,44,45,50)(39,48,42,49);(19,23)(20,28)(21,31)(22,34)(24,36)(25,35)(26,39)(27,40)(29,42)(30,44)(32,45)(33,46)(37,47)(38,48)(41,49)(43,50)140 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10493420);