Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #471 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C28×C4 |
| direct product | C2×#21075 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | #121 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| socle | C23 | C2×#121 |
| prime radical | C23×#41 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3222 conjugacy classes of subgroups.
| minimal degree | μ(G) = 38 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 38 points:
(1,2);(3,4,6,5)(7,8)(9,18)(10,23)(11,16)(12,27)(13,17)(14,21)(15,26)(19,33)(20,29)(22,32)(24,38)(25,30)(28,34)(31,36)(35,37);(3,5,6,4)(7,9,12,21,13,22,11,20)(8,14,16,18,17,29,27,32)(10,19,25,35,26,36,24,34)(15,28,30,31,23,37,38,33);(7,10,12,25,13,26,11,24)(8,15,16,30,17,23,27,38)(9,19,21,35,22,36,20,34)(14,28,18,31,29,37,32,33);(7,11,13,12)(8,16,17,27)(9,20,22,21)(10,24,26,25)(14,18,29,32)(15,30,23,38)(19,34,36,35)(28,31,37,33)128 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6278861);