Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D8 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C16 · Schur multiplier (H2) = C25 |
| direct product | C2×C16×SD16 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C16 | D4 |
| commutator | C4 | C23×C16 |
| Frattini, cosocle | C4×C8 | C24 |
| socle | C23 | C8×D4 |
| prime radical | C22×D4 | C2×C8 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C16 ◁ C2×C4×C16 ◁ G | |
| LECS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
546 conjugacy classes of subgroups.
| minimal degree | μ(G) = 26 |
| transitive degree | μt(G) = 256 |
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);(20,22)(21,25)(24,26)224 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 390712);