Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #1347 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C24 |
| direct product | C2×#7261 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | #102 |
| commutator | C2×C8 | C2×C42 |
| Frattini, cosocle | C23×C8 | C23 |
| socle | C23 | #340 |
| prime radical | C25 | Q16 |
| LCS | G ▷ C2×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C24×C4 ◁ C24×C8 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
846 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)(11,12,15,20,17,21,16,31)(13,22,24,38,25,39,32,36)(14,26,28,37,30,41,29,19)(18,27,33,40,35,42,34,23);(4,7,9,8)(11,13,15,24,17,25,16,32)(12,18,20,33,21,35,31,34)(14,27,28,40,30,42,29,23)(19,36,26,22,37,38,41,39)116 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 34715);