Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #25443 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
| direct product | C2×#28941 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | D42 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#537 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2180 conjugacy classes of subgroups.
| minimal degree | μ(G) = 42 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 42 points:
(1,2);(3,4)(5,10)(6,8)(7,9)(11,12)(13,23)(14,27)(15,18)(16,21)(17,22)(19,24)(20,29)(25,39)(26,32)(28,33)(30,36)(31,34)(35,41)(37,38)(40,42);(3,5,7,6)(4,8,9,10)(12,18,22,21)(13,24,26,31)(14,28)(19,34)(20,27,36,33)(23,32)(25,40,41,38)(29,30);(3,6,7,5)(4,8,9,10)(11,13,15,24,17,26,16,31)(12,19,21,23,22,34,18,32)(14,25,29,38,28,41,30,40)(20,35,27,37,36,39,33,42);(4,9)(8,10)(11,14,16,30,17,28,15,29)(12,20,18,33,22,36,21,27)(13,25,31,40,26,41,24,38)(19,35,32,42,34,39,23,37)80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6367240);