Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #29748 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
| direct product | C2×#29888 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#121 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1668 conjugacy classes of subgroups.
| minimal degree | μ(G) = 50 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 50 points:
(1,2);(3,4)(5,9)(6,10)(7,8)(11,17)(12,15)(13,18)(14,16)(19,20,24,29,23,28,25,30)(21,31,34,46,33,26,40,42)(22,35,38,44,37,45,39,27)(32,48,36,41,49,43,50,47);(4,8)(9,15)(10,16)(17,18)(19,21)(20,26)(22,36)(23,33)(24,34)(25,40)(27,43)(28,31)(29,42)(30,46)(32,39)(35,47)(37,50)(38,49)(41,45)(44,48);(3,5,7,12)(4,9,8,15)(6,13,14,11)(10,18,16,17)(19,22,23,37)(20,27,28,44)(21,32,33,49)(24,39,25,38)(26,41,31,47)(29,45,30,35)(34,50,40,36)(42,48,46,43);(3,6,7,14)(4,10,8,16)(5,11,12,13)(9,17,15,18)(22,37)(27,44)(32,49)(35,45)(36,50)(38,39)(41,47)(43,48)80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6343407);