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×#13172 |
| 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)(11,24)(15,28)(16,32)(17,23)(19,25)(20,34)(21,33)(22,35)(26,30)(31,38)(36,37);(3,5,6,4)(7,8)(9,15)(10,23)(11,16)(12,14)(13,17)(18,30)(19,29)(20,36)(21,26)(22,31)(24,27)(25,32)(28,34)(33,38)(35,37);(7,9,19,33,12,21,11,18,13,22,24,34,10,20,25,35)(8,15,29,38,14,26,16,30,17,31,27,28,23,36,32,37)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 402233);