Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33764 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
| direct product | C2×#33770 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D8 |
| commutator, Frattini, cosocle | C2×C8 | C25 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×#41 ◁ C22×#131 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
1654 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)(5,9)(6,8)(7,10)(11,12,15,21)(13,24,26,18)(14,23,17,20)(16,31,30,34)(19,37,38,27)(22,29,40,33)(25,35,28,39)(32,41,42,36);(3,5)(4,7)(6,10)(8,9)(11,13,15,26)(12,18,21,24)(14,28,17,25)(16,27,30,37)(19,31,38,34)(20,39,23,35)(22,36,40,41)(29,42,33,32);(5,10)(7,9)(11,13)(12,19)(14,25)(15,26)(16,32)(17,28)(18,34)(20,36)(21,38)(22,39)(23,41)(24,31)(27,33)(29,37)(30,42)(35,40);(4,8)(7,9)(11,14,15,17)(12,20,21,23)(13,25,26,28)(16,29,30,33)(18,35,24,39)(19,36,38,41)(22,34,40,31)(27,32,37,42)80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7530499);