Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3026 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C211 |
| direct product | C2×#20962 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#120 |
| commutator | C23 | C24×C4 |
| Frattini, cosocle | C24 | C25 |
| prime radical | C23×#41 | C22 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C24 ▷ C2 ▷ C1 | |
6250 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);(4,6)(7,8)(9,13)(10,20,22,14)(11,23)(12,16)(15,24)(17,25,30,33)(18,31)(19,27)(21,28,35,34)(26,32)(29,37,38,36);(3,4,5,6)(7,9,12,19)(8,13,16,27)(10,17,22,30)(11,18,24,32)(14,25,20,33)(15,26,23,31)(21,29,35,38)(28,36,34,37);(7,10)(8,14)(9,17)(11,21)(12,22)(13,25)(15,28)(16,20)(18,29)(19,30)(23,34)(24,35)(26,36)(27,33)(31,37)(32,38);(7,11,12,24)(8,15,16,23)(9,18,19,32)(10,21,22,35)(13,26,27,31)(14,28,20,34)(17,29,30,38)(25,36,33,37)116 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6278660);