Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12503 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C8 |
| direct product | C2×#12516 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23 | #493 |
| commutator, Frattini, cosocle | C4×C8 | C24 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C42 ◁ C2×C82 ◁ G | |
| LECS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
730 conjugacy classes of subgroups.
| minimal degree | μ(G) = 42 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 4 generators on 42 points:
(1,2);(4,7)(5,8)(9,10)(11,12,16,20)(13,21,25,37)(14,26,28,41)(15,29,27,19)(17,30,32,23)(18,33,34,22)(24,40,39,36)(31,38,35,42);(11,13,22,38,15,24,28,30,16,25,33,42,27,39,14,23)(12,17,26,36,19,31,34,37,20,32,41,40,29,35,18,21);(3,4,8,10,6,9,5,7)140 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 398658);