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×#60338 |
| 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)(19,20,23,28,24,26,35,39)(21,25,31,38,29,36,43,47)(22,32,33,46,34,27,40,41)(30,44,45,50,42,37,48,49);(3,5,7,13)(4,8,10,11)(6,14,15,12)(9,17,18,16)(19,21)(20,25)(22,30)(23,31)(24,29)(26,36)(27,37)(28,38)(32,44)(33,45)(34,42)(35,43)(39,47)(40,48)(41,49)(46,50);(3,6,7,15)(4,9,10,18)(5,12,13,14)(8,16,11,17)(20,26)(21,29)(27,32)(28,39)(30,42)(31,43)(41,46)(45,48);(4,10)(8,11)(9,18)(16,17)(21,29)(25,36)(30,42)(31,43)(37,44)(38,47)(45,48)(49,50);(19,22,23,33,24,34,35,40)(20,27,28,41,26,32,39,46)(21,30,31,45,29,42,43,48)(25,37,38,49,36,44,47,50)152 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10481743);