Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3484 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C4 · Schur multiplier (H2) = C213 |
| direct product | C2×#60212 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C25 |
| commutator | C22 | C25×C4 |
| Frattini, cosocle | C2×C4 | C26 |
| socle | C23 | C24×C4 |
| prime radical | C2×#3455 | C2 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
6814 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)(5,8)(6,7)(9,10)(11,12,15,19,16,20,26,32)(13,17,23,29,22,28,35,39)(14,18,25,31,24,30,36,40)(21,27,34,38,33,37,41,42);(4,7)(8,10)(11,13)(12,17)(14,21)(15,23)(16,22)(18,27)(19,29)(20,28)(24,33)(25,34)(26,35)(30,37)(31,38)(32,39)(36,41)(40,42);(11,14,15,25,16,24,26,36)(12,18,19,31,20,30,32,40)(13,21,23,34,22,33,35,41)(17,27,29,38,28,37,39,42);(3,5,6,9)(4,8,7,10)(13,22)(17,28)(21,33)(23,35)(27,37)(29,39)(34,41)(38,42);(14,24)(18,30)(21,33)(25,36)(27,37)(31,40)(34,41)(38,42)176 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10481438);