Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #8325 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C29 |
| direct product | C2×#10985 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #379 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C23×#41 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
4374 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);(7,8)(9,17)(10,21,23,14)(11,24)(12,16)(13,19)(15,26)(18,32,34,31)(20,30)(22,29,38,36)(25,28)(27,33,35,37);(3,4)(5,6)(7,9,12,20)(8,13,16,28)(10,18,23,34)(11,25,26,19)(14,27,21,35)(15,30,24,17)(22,37,38,33)(29,32,36,31);(4,6)(7,10)(8,14)(9,18)(11,22)(12,23)(13,27)(15,29)(16,21)(17,31)(19,33)(20,34)(24,36)(25,37)(26,38)(28,35)(30,32)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 264665);