Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3532 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C4 · Schur multiplier (H2) = C213 |
| direct product | C2×#60346 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C25 |
| commutator | C22 | C25×C4 |
| Frattini, cosocle | C2×C4 | C26 |
| socle | C23 | C24×C4 |
| prime radical | C2×#3531 | C2 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
5946 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)(19,20,23,28,24,26,35,40)(21,25,31,38,29,36,43,47)(22,27,33,42,34,39,46,49)(30,37,44,48,32,41,45,50);(3,5,7,13)(4,8,10,11)(6,14,15,12)(9,17,18,16)(19,21)(20,25)(22,32)(23,31)(24,29)(26,36)(27,41)(28,38)(30,34)(33,45)(35,43)(37,39)(40,47)(42,50)(44,46)(48,49);(3,6,7,15)(4,9,10,18)(5,12,13,14)(8,16,11,17)(20,26)(21,29)(27,39)(28,40)(30,32)(31,43)(42,49)(44,45);(4,10)(8,11)(9,18)(16,17)(21,29)(25,36)(30,32)(31,43)(37,41)(38,47)(44,45)(48,50);(19,22,23,33,24,34,35,46)(20,27,28,42,26,39,40,49)(21,30,31,44,29,32,43,45)(25,37,38,48,36,41,47,50)152 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10481751);