Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #29602 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
| direct product | C2×#30052 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C24 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C2×#376 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2026 conjugacy classes of subgroups.
| minimal degree | μ(G) = 50 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 50 points:
(1,2);(3,4,6,7)(5,8,9,10)(11,12)(13,16)(14,15)(17,18)(19,20)(21,25)(22,32)(23,27)(24,28)(26,34)(29,47)(30,40)(31,41)(33,50)(35,46)(36,44)(37,45)(38,43)(39,48)(42,49);(4,7)(8,10)(19,21,23,30)(20,25,27,40)(22,33,34,49)(24,37,36,31)(26,42,32,50)(28,45,44,41)(29,38,48,35)(39,46,47,43);(3,5)(4,8)(6,9)(7,10)(12,15)(16,18)(19,22,23,34)(20,26,27,32)(21,29,30,48)(24,38,36,35)(25,39,40,47)(28,46,44,43)(31,49,37,33)(41,50,45,42);(5,9)(8,10)(11,13,14,17)(12,16,15,18)(20,27)(25,40)(26,32)(28,44)(39,47)(41,45)(42,50)(43,46)80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6341342);