Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #29748 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
| direct product | C2×#29860 |
| 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 | |
1950 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);(3,4,6,7)(5,8,9,10)(11,12,17,22,15,20,16,21)(13,23,32,34,25,18,26,38)(14,27,31,19,29,37,30,36)(24,40,42,39,41,35,28,33);(4,7)(8,10)(11,13)(12,18)(14,28)(15,25)(16,26)(17,32)(19,35)(20,23)(21,34)(22,38)(24,31)(27,39)(29,42)(30,41)(33,37)(36,40);(5,9)(8,10)(11,14)(12,19)(13,24)(15,29)(16,31)(17,30)(18,33)(20,36)(21,37)(22,27)(23,39)(25,41)(26,42)(28,32)(34,40)(35,38);(3,5,6,9)(4,8,7,10)(14,29)(19,36)(24,41)(27,37)(28,42)(30,31)(33,39)(35,40)80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6342778);