Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24468 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28×C4 |
| direct product | C2×#25594 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#120 |
| 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 | |
3386 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)(5,9)(6,7)(8,10)(11,12,16,20)(13,17,22,31)(14,18,25,34)(15,27,29,19)(21,37,38,30)(23,36,28,32)(24,33,39,42)(26,41,40,35);(4,7)(8,9)(11,13)(12,17)(14,24)(15,28)(16,22)(18,33)(19,36)(20,31)(21,26)(23,29)(25,39)(27,32)(30,35)(34,42)(37,41)(38,40);(11,14)(12,18)(13,21,22,38)(15,26)(16,25)(17,30,31,37)(19,35)(20,34)(23,24,28,39)(27,41)(29,40)(32,33,36,42);(3,5,6,10)(4,8,7,9)(14,25)(18,34)(21,38)(24,39)(26,40)(30,37)(33,42)(35,41)80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6356283);