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×#13646 |
| 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 | |
466 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)(7,8,11,16,12,17,25,32,13,9,18,20,26,21,33,35,10,14,22,27,23,28,36,37,24,19,15,29,30,34,31,38);(4,6)(8,14)(9,19)(16,27)(17,28)(20,29)(21,34)(32,37)(35,38);(7,9,11,20,12,21,25,35,13,14,18,27,26,28,33,37,10,19,22,29,23,34,36,38,24,8,15,16,30,17,31,32)224 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 419937);