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×#60327 |
| 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)(11,12,15,19,16,20,27,34)(13,17,23,30,21,28,35,39)(14,18,26,33,25,32,38,42)(22,29,36,40,24,31,37,41);(3,5)(4,7)(6,10)(8,9)(11,13)(12,17)(14,24)(15,23)(16,21)(18,31)(19,30)(20,28)(22,25)(26,37)(27,35)(29,32)(33,41)(34,39)(36,38)(40,42);(5,10)(7,9)(13,21)(14,25)(17,28)(18,32)(23,35)(26,38)(30,39)(33,42);(4,8)(7,9)(13,21)(17,28)(22,24)(23,35)(29,31)(30,39)(36,37)(40,41);(11,14,15,26,16,25,27,38)(12,18,19,33,20,32,34,42)(13,22,23,36,21,24,35,37)(17,29,30,40,28,31,39,41)152 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10481683);