Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24390 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28×C4 |
| direct product | C2×#28456 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C2×#376 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3180 conjugacy classes of subgroups.
| minimal degree | μ(G) = 42 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 42 points:
(1,2);(4,5)(11,12,15,19,16,20,29,36)(13,17,23,32,22,31,39,42)(14,24,26,34,27,35,28,18)(21,37,38,33,25,41,40,30);(3,4,6,5)(7,8,9,10)(11,13)(12,17)(14,25)(15,23)(16,22)(18,33)(19,32)(20,31)(21,27)(24,41)(26,40)(28,38)(29,39)(30,34)(35,37)(36,42);(8,10)(11,14,16,27)(12,18,20,34)(13,21,22,25)(15,28,29,26)(17,30,31,33)(19,35,36,24)(23,40,39,38)(32,41,42,37);(3,5,6,4)(13,22)(17,31)(21,25)(23,39)(30,33)(32,42)(37,41)(38,40)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6372735);