Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3198 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C213 |
| direct product | C2×#60957 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C22×D4 |
| commutator, Frattini, cosocle | C2×C4 | C26 |
| socle | C23 | C23×D4 |
| prime radical | C2×#3203 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C2×C4×#41 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
6130 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)(6,10)(7,9)(8,12)(13,17)(14,16)(15,18)(19,20,24,28)(21,25,30,39)(22,32,34,42)(23,35,37,27)(26,36,41,33)(29,43,44,50)(31,47,46,40)(38,48,49,45);(3,5)(4,8)(6,13)(7,12)(9,11)(10,15)(14,18)(16,17)(19,21)(20,25)(22,29)(23,31)(24,30)(26,38)(27,40)(28,39)(32,43)(33,45)(34,44)(35,47)(36,48)(37,46)(41,49)(42,50);(5,12)(8,11)(13,18)(15,17)(19,22,24,34)(20,26,28,41)(21,29,30,44)(23,36,37,33)(25,38,39,49)(27,42,35,32)(31,48,46,45)(40,50,47,43);(4,9)(8,11)(10,16)(15,17)(21,30)(25,39)(29,44)(31,46)(38,49)(40,47)(43,50)(45,48);(3,6,7,14)(4,10,9,16)(5,13,12,18)(8,15,11,17)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10482993);