Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #353 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C8 · Schur multiplier (H2) = C24 |
| direct product | C2×#7194 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | #99 |
| commutator | C2×C4 | C2×C4×C8 |
| Frattini, cosocle | C22×M16 | C23 |
| socle | C23 | #323 |
| prime radical | C25 | M16 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C23×C4 ◁ C23×M16 ◁ G | |
| LECS | G ▷ C22×M16 ▷ C2×C4 ▷ C2 ▷ C1 | |
818 conjugacy classes of subgroups.
| minimal degree | μ(G) = 38 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 38 points:
(1,2);(7,8,11,16,12,15,26,31,13,17,27,33,10,23,24,32)(9,18,20,34,21,30,36,38,22,35,25,37,19,14,28,29);(3,4,5,6)(7,9,10,19,13,22,12,21)(8,14,15,18,17,30,23,35)(11,25,24,36,27,20,26,28)(16,29,31,34,33,38,32,37)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 34121);