Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24802 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
| direct product | C2×#28591 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#121 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#525 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2348 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)(6,8)(7,10)(11,12)(13,18)(14,26)(15,30)(16,20)(17,21)(19,28)(22,38)(23,40)(24,33)(25,34)(27,32)(29,36)(31,37)(35,39)(41,42);(3,5,6,10)(4,7,8,9)(11,13,16,24,17,25,15,23)(12,18,20,33,21,34,30,40)(14,27,28,39,29,41,31,22)(19,35,36,42,37,38,26,32);(11,14,15,28,17,29,16,31)(12,19,20,26,21,37,30,36)(13,22,23,27,25,39,24,41)(18,32,33,38,34,42,40,35);(4,8)(7,9)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6366787);