Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1812 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C26×C4 |
| direct product | C2×#10867 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | #117 |
| commutator | C2×C4 | C22×C42 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C23 | #380 |
| prime radical | C25 | C2×Q8 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C23×M16 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2082 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);(7,8,12,17,13,16,11,26)(9,18,20,29,21,14,27,30)(10,22,24,31,25,37,23,15)(19,33,34,32,35,36,38,28);(3,4,5,6)(7,9)(8,14)(10,19)(11,20)(12,27)(13,21)(15,28)(16,18)(17,29)(22,36)(23,34)(24,38)(25,35)(26,30)(31,32)(33,37);(7,10,11,23,13,25,12,24)(8,15,17,22,16,31,26,37)(9,19,20,34,21,35,27,38)(14,28,29,36,18,32,30,33)128 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 260859);