Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 1 |
| isoclinism | isoclinic class of C1 |
| automorphism | Aut(G) order 12582912 |
| homology | Abelianization (H1) = G · Schur multiplier (H2) = C23×C43 |
| abelian | C2×C42×C16 |
| multiplicative | (ℤ/1360)ˣ |
| subgroup | G/H | |
|---|---|---|
| center | G | C1 |
| commutator | C1 | G |
| Frattini, cosocle | C22×C8 | C24 |
| socle, prime radical | C24 | C22×C8 |
| LCS | G ▷ C1 | |
| UCS | C1 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
1410 conjugacy classes of subgroups.
| minimal degree | μ(G) = 26 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 4 generators on 26 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16);(17,18,19,20);(21,22,23,24);(25,26)512 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 400018);