Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32944 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
| direct product | C2×#33065 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#121 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2186 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)(13,20,27,34)(14,19,26,35)(15,29,32,37)(16,23,17,22)(21,31,38,30)(25,39,41,33)(28,36,40,42);(4,7)(8,10)(11,13,16,14,18,27,17,26)(12,19,22,20,24,35,23,34)(15,25,30,28,32,41,31,40)(21,33,29,36,38,39,37,42);(3,5,6,9)(4,8,7,10)(11,14,16,27,18,26,17,13)(12,20,22,35,24,34,23,19)(15,28,30,41,32,40,31,25)(21,36,29,39,38,42,37,33);(11,15)(12,21)(13,25)(14,28)(16,30)(17,31)(18,32)(19,33)(20,36)(22,29)(23,37)(24,38)(26,40)(27,41)(34,42)(35,39)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6536726);