Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9758 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C8 |
| direct product | C2×#9760 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #390 |
| commutator | C42 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| prime radical | C2×#3147 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
1298 conjugacy classes of subgroups.
| minimal degree | μ(G) = 42 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 42 points:
(1,2);(3,4)(5,9)(6,10)(7,8)(11,12)(13,17,22,32)(14,24)(15,29)(16,19)(18,27)(20,26)(21,35,25,39)(23,38,36,31)(28,40)(30,41,33,37)(34,42);(3,5,9,4,7,10,6,8)(11,13,16,22)(12,17)(14,25,28,21)(15,23,26,36)(18,33)(19,32)(20,31)(24,39)(27,41,42,37)(29,38)(30,34)(35,40);(4,8)(5,10)(11,14,26,42,16,28,15,27)(12,18,29,40,19,34,20,24)(13,21,36,41,22,25,23,37)(17,30,38,39,32,33,31,35)80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 112737);