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×#60318 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C25 |
| commutator | C22 | C25×C4 |
| Frattini, cosocle | C2×C4 | C26 |
| socle | C23 | C24×C4 |
| prime radical | C2×#3529 | C2 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
6586 conjugacy classes of subgroups.
| minimal degree | μ(G) = 42 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 42 points:
(1,2);(3,4)(5,9)(6,8)(7,10)(11,12,15,20,16,21,28,36)(13,17,24,30,22,18,31,33)(14,25,26,39,27,19,34,35)(23,32,38,42,37,29,40,41);(3,5)(4,7)(6,10)(8,9)(11,13)(12,17)(14,23)(15,24)(16,22)(18,21)(19,29)(20,30)(25,32)(26,38)(27,37)(28,31)(33,36)(34,40)(35,41)(39,42);(5,10)(7,9)(11,13)(12,18)(14,23)(15,24)(16,22)(17,21)(19,32)(20,33)(25,29)(26,38)(27,37)(28,31)(30,36)(34,40)(35,42)(39,41);(4,8)(7,9)(13,22)(17,18)(23,37)(24,31)(29,32)(30,33)(38,40)(41,42);(11,14,15,26,16,27,28,34)(12,19,20,35,21,25,36,39)(13,23,24,38,22,37,31,40)(17,29,30,41,18,32,33,42)152 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10481674);