Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #1343 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C24 |
| direct product | C2×#7253 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | #99 |
| commutator | C22×C4 | C2×C42 |
| Frattini, cosocle | C2×#117 | C23 |
| socle | C23 | #342 |
| prime radical | C25 | C4⋊C4 |
| LCS | G ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C23×C4 ◁ C22×#117 ◁ G | |
| LECS | G ▷ C2×#117 ▷ C23 ▷ C2 ▷ C1 | |
856 conjugacy classes of subgroups.
| minimal degree | μ(G) = 42 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 42 points:
(1,2);(3,4)(5,7)(6,9)(8,10)(11,12,15,20,17,22,16,21)(13,23,25,40,27,36,26,41)(14,19,29,37,30,38,31,39)(18,32,33,28,35,42,34,24);(4,7,9,8)(11,13,16,26,17,27,15,25)(12,18,21,34,22,35,20,33)(14,28,29,42,30,24,31,32)(19,36,37,41,38,23,39,40)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 34697);