Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2010 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
| direct product | C2×#11374 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | #127 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C23 | #399 |
| prime radical | C25 | #41 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1670 conjugacy classes of subgroups.
| minimal degree | μ(G) = 38 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 38 points:
(1,2);(3,4)(5,6)(7,8)(9,18,22,34)(10,23)(11,16)(12,17)(13,27)(14,20,30,21)(15,26)(19,35,36,31)(24,38,37,29)(25,33)(28,32);(4,6)(7,9,13,20,12,22,11,21)(8,14,27,18,17,30,16,34)(10,24,28,36,26,37,25,19)(15,31,33,29,23,35,32,38);(4,6)(7,10,13,28,12,26,11,25)(8,15,27,33,17,23,16,32)(9,19,20,24,22,36,21,37)(14,29,18,35,30,38,34,31)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 265779);