Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33605 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
| direct product | C2×#33705 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D8 |
| commutator, Frattini, cosocle | C2×C8 | C25 |
| prime radical | C24 | C2×D8 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
1662 conjugacy classes of subgroups.
| minimal degree | μ(G) = 42 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 42 points:
(1,2);(3,4)(5,8)(6,7)(9,10)(11,12,18,24)(13,20,25,37)(14,19,28,34)(15,22,31,40)(16,21,32,39)(17,33,30,23)(26,38,41,36)(27,42,29,35);(4,7)(8,10)(11,13,16,26,17,27,31,28,18,25,32,41,30,29,15,14)(12,19,22,35,23,36,39,37,24,34,40,42,33,38,21,20);(3,5,6,9)(4,8,7,10)(11,14,15,29,30,41,32,25,18,28,31,27,17,26,16,13)(12,20,21,38,33,42,40,34,24,37,39,36,23,35,22,19);(13,25)(14,28)(19,34)(20,37)(26,41)(27,29)(35,42)(36,38)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7530365);