Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32941 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
| direct product | C2×#33062 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#121 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C22×AGL(1,ℤ/8) | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2240 conjugacy classes of subgroups.
| minimal degree | μ(G) = 42 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 42 points:
(1,2);(3,4)(5,8)(6,7)(9,10)(11,12)(13,23)(14,18)(15,29)(16,33)(17,22)(19,26)(20,31)(21,24)(25,34)(27,36)(28,35)(30,38)(32,37)(39,40)(41,42);(4,7)(8,10)(11,13,24,28,17,26,16,14)(12,18,33,19,22,35,21,23)(15,25,31,40,32,41,30,27)(20,34,29,36,38,42,37,39);(3,5,6,9)(4,8,7,10)(11,14,16,26,17,28,24,13)(12,19,21,18,22,23,33,35)(15,27,30,41,32,40,31,25)(20,36,37,34,38,39,29,42);(11,15)(12,20)(13,25)(14,27)(16,30)(17,32)(18,34)(19,36)(21,37)(22,38)(23,39)(24,31)(26,41)(28,40)(29,33)(35,42)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6536722);