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×#62891 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C26 |
| commutator, Frattini, cosocle | C22 | C27 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C22 ▷ C1 | |
31048 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);(3,4,8,9)(5,10,12,7)(6,14,16,11)(13,15,18,17)(19,20,25,30)(21,31,35,26)(22,23,28,27)(24,40,41,29)(32,33,36,38)(34,42,48,45)(37,43,44,39)(46,50,49,47);(4,9)(5,12)(11,14)(13,18)(19,21,25,35)(20,26,30,31)(22,32,28,36)(23,38,27,33)(24,34,41,48)(29,42,40,45)(37,46,44,49)(39,50,43,47);(3,5)(4,10)(6,13)(7,9)(8,12)(11,17)(14,15)(16,18)(19,21)(20,26)(22,36)(23,33)(24,34)(25,35)(27,38)(28,32)(29,42)(30,31)(37,49)(39,47)(40,45)(41,48)(43,50)(44,46);(3,6,8,16)(4,11,9,14)(5,13,12,18)(7,15,10,17)(19,22,25,28)(20,27,30,23)(21,32,35,36)(24,37,41,44)(26,33,31,38)(29,43,40,39)(34,46,48,49)(42,47,45,50);(19,23,25,27)(20,28,30,22)(21,33,35,38)(24,39,41,43)(26,32,31,36)(29,44,40,37)(34,47,48,50)(42,46,45,49);(3,7,8,10)(4,5,9,12)(6,15,16,17)(11,13,14,18)(19,24,25,41)(20,29,30,40)(21,34,35,48)(22,37,28,44)(23,39,27,43)(26,42,31,45)(32,46,36,49)(33,47,38,50)140 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10493422);