Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2230 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
| direct product | C2×#13273 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #476 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C22×C8 | C24 |
| prime radical | C24 | AGL(1,ℤ/8) |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
854 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);(4,6)(7,8,12,16,11,28,27,38,14,19,26,36,13,18,10,17)(9,20,24,15,30,33,29,32,25,31,21,35,23,37,22,34);(3,4,5,6)(7,9,11,23,14,25,13,30)(8,15,18,32,19,35,28,34)(10,21,12,29,27,24,26,22)(16,31,17,37,36,20,38,33);(7,10,13,26,14,27,11,12)(8,16,28,38,19,36,18,17)(9,21,30,22,25,24,23,29)(15,31,34,33,35,20,32,37)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 402543);