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×#62380 |
| 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)(21,26,32,42)(22,33,36,47)(23,28,24,38)(27,34,43,35)(30,44,46,49)(31,40,37,41)(39,48,50,45);(3,5,7,13)(4,8,10,11)(6,14,15,12)(9,17,18,16)(19,21)(20,26)(22,30)(23,37)(24,31)(25,32)(27,39)(28,41)(29,42)(33,44)(34,48)(35,45)(36,46)(38,40)(43,50)(47,49);(19,22,23,34,25,36,24,35)(20,27,38,47,29,43,28,33)(21,30,37,48,32,46,31,45)(26,39,40,49,42,50,41,44);(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)(20,28,29,38)(21,31,32,37)(22,34,36,35)(26,40,42,41)(27,33,43,47)(30,45,46,48)(39,49,50,44)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10485616);