Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #62265 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C213 |
| direct product | C2×#62385 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C2×C4 | C26 |
| prime radical | C2×#466 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#533 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
5892 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)(5,11)(6,9)(7,10)(8,13)(12,17)(14,16)(15,18)(19,20,25,29)(21,26,32,43)(22,33,37,49)(23,28,24,39)(27,35,45,36)(30,46,34,48)(31,41,38,42)(40,50,44,47);(3,5,7,13)(4,8,10,11)(6,14,15,12)(9,17,18,16)(19,21)(20,26)(22,34)(23,38)(24,31)(25,32)(27,44)(28,42)(29,43)(30,37)(33,48)(35,47)(36,50)(39,41)(40,45)(46,49);(19,22,24,36,25,37,23,35)(20,27,28,33,29,45,39,49)(21,30,31,47,32,34,38,50)(26,40,42,46,43,44,41,48);(3,6,7,15)(4,9,10,18)(5,12,13,14)(8,16,11,17);(4,10)(8,11)(9,18)(16,17)(19,23,25,24)(20,28,29,39)(21,31,32,38)(22,35,37,36)(26,41,43,42)(27,33,45,49)(30,47,34,50)(40,48,44,46)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10485622);