Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of AΣL(1,𝔽16) |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C8 · Schur multiplier (H2) = C23×C4 |
| direct product | C2×#7188 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | #99 |
| commutator | C23 | C2×C4×C8 |
| Frattini, cosocle | C2×#117 | C23 |
| socle | C23 | #342 |
| prime radical | C25 | C42 |
| LCS | G ▷ C23 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C23×C4 ◁ C22×#117 ◁ G | |
| LECS | G ▷ C2×#117 ▷ C23 ▷ C2 ▷ C1 | |
842 conjugacy classes of subgroups.
| minimal degree | μ(G) = 42 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 42 points:
(1,2);(3,4,5,7,6,8,9,10)(11,12)(13,21)(14,25,26,18)(15,19)(16,20)(17,22,29,35)(23,33)(24,34)(27,38,39,31)(28,32)(30,36,40,42)(37,41);(11,13,15,23,16,24,28,37)(12,17,19,30,20,29,32,40)(14,22,27,36,26,35,39,42)(18,21,31,33,25,34,38,41)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 34107);