Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9755 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
| direct product | C2×#9763 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #390 |
| commutator | C42 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| prime radical | C24 | #121 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
1076 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);(3,4,5,7,6,8,10,9)(11,12)(13,22,25,33)(14,26)(15,20)(16,31)(17,21)(18,24,34,30)(19,28)(23,32,38,42)(27,35)(29,40)(36,41,39,37);(4,7,8,9)(11,13,15,30,17,25,16,24)(12,18,20,33,21,34,31,22)(14,23,27,41,29,38,28,37)(19,32,26,39,35,42,40,36);(4,8)(7,9)(11,14,16,28,17,29,15,27)(12,19,20,26,21,35,31,40)(13,23,24,37,25,38,30,41)(18,32,33,39,34,42,22,36)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 112773);