Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1462 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C26×C4 |
| direct product | C2×#10803 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C2×C22⋊C4 |
| commutator | C23 | C22×C42 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C23 | C4×C22⋊C4 |
| prime radical | C25 | C22×C4 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C24×C8 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1834 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);(4,5)(7,8)(9,13)(10,20,21,14)(11,16)(12,15)(17,29,30,24)(18,26)(19,25)(22,33,34,27)(23,28)(31,37,38,35)(32,36);(3,4,6,5)(7,9,11,18,12,19,23,32)(8,13,16,26,15,25,28,36)(10,17,22,31,21,30,34,38)(14,24,27,35,20,29,33,37);(3,5,6,4)(7,10)(8,14)(9,17)(11,22)(12,21)(13,24)(15,20)(16,27)(18,31)(19,30)(23,34)(25,29)(26,35)(28,33)(32,38)(36,37)128 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 260690);