Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24292 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
| direct product | C2×#25437 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | D42 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C22×AGL(1,ℤ/8) | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2684 conjugacy classes of subgroups.
| minimal degree | μ(G) = 38 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 38 points:
(1,2);(3,4,6,5)(7,8,12,18)(9,14,22,31)(10,23,26,34)(11,17,27,16)(13,21,29,20)(15,32,35,25)(19,36,24,28)(30,38,33,37);(3,5,6,4)(7,9,11,20,12,22,27,21)(8,13,16,14,18,29,17,31)(10,19,25,37,26,24,32,38)(15,28,23,30,35,36,34,33);(4,5)(7,9)(8,14)(10,24)(11,20)(12,22)(13,17)(15,33)(16,29)(18,31)(19,26)(21,27)(23,28)(25,38)(30,35)(32,37)(34,36);(7,10,11,25,12,26,27,32)(8,15,16,23,18,35,17,34)(9,19,20,37,22,24,21,38)(13,28,14,30,29,36,31,33)80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6351491);