Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #8603 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
| direct product | C2×#11259 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | #117 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C23 | #382 |
| prime radical | C25 | #41 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C22×#117 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1654 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)(9,17,20,13)(10,21,23,36)(11,16)(12,15)(14,22,27,37)(18,31)(19,30,34,26)(24,28)(25,32)(29,38)(33,35);(3,4)(5,6)(7,9,11,19,12,20,24,34)(8,13,16,26,15,17,28,30)(10,18,22,32,23,33,37,38)(14,25,21,35,27,29,36,31);(4,6)(7,10)(8,14,15,27)(9,18)(11,22)(12,23)(13,25,17,29)(16,21,28,36)(19,32)(20,33)(24,37)(26,35,30,31)(34,38)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 265463);