Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of H(3,ℤ/4) |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C82 · Schur multiplier (H2) = C23 |
| direct product | C2×#7330 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C8 | C42 |
| commutator | C4 | C2×C82 |
| Frattini, cosocle | C4×C16 | C23 |
| socle, prime radical | C23 | #346 |
| LCS | G ▷ C4 ▷ C1 | |
| UCS | C1 ◁ C22×C8 ◁ G | |
| LECS | G ▷ C4×C16 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
210 conjugacy classes of subgroups.
| minimal degree | μ(G) = 42 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 3 generators on 42 points:
(1,2);(11,12,13,18,14,20,23,33,15,21,24,31,25,35,37,41,16,19,22,32,26,34,36,42,27,17,28,29,38,30,39,40);(3,4,5,7,6,8,9,10)(12,17,19,21)(13,22)(18,31,32,29)(20,30,34,35)(23,36)(24,28)(33,41,42,40)(37,39)176 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 56690);