Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2221 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
| direct product | C2×#13171 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #466 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C22×C8 | C24 |
| prime radical | C24 | #135 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
650 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);(3,4,6,5)(8,14)(9,18)(10,12)(11,19)(15,28)(16,31)(17,23)(20,33)(21,32)(22,34)(24,25)(26,29)(30,38)(35,37);(3,5,6,4)(7,8)(9,15)(10,23)(11,16)(12,14)(13,17)(18,29)(19,27)(20,35)(21,26)(22,30)(24,36)(25,31)(28,33)(32,38)(34,37);(7,9,19,33,12,21,25,34,13,22,24,32,10,20,11,18)(8,15,27,28,14,26,31,37,17,30,36,38,23,35,16,29)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 402232);