Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #120 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C16 · Schur multiplier (H2) = C25 |
| direct product | C2×#13645 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C16 | C23 |
| commutator | C22 | C23×C16 |
| Frattini, cosocle | C2×C16 | C24 |
| socle | C23 | C22×C16 |
| prime radical | C22×D4 | C16 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C16 ◁ G | |
| LECS | G ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
472 conjugacy classes of subgroups.
| minimal degree | μ(G) = 38 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 38 points:
(1,2);(3,4)(5,6)(7,8,10,15,11,16,22,30,12,9,18,19,24,20,33,34,13,17,23,31,25,32,36,38,26,21,14,27,28,35,29,37);(4,6);(7,9,10,19,11,20,22,34,12,17,18,31,24,32,33,38,13,21,23,27,25,35,36,37,26,8,14,15,28,16,29,30)224 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 419936);