Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of AGL(1,ℤ/8) |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C43 · Schur multiplier (H2) = C25×C42 |
| direct product | C4×#1738 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C42 | C2×D4 |
| commutator | C4 | C2×C43 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C23 | C22×C4⋊C4 |
| prime radical | C25 | C2×Q8 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C42 ◁ C23×C42 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2534 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2,3,4);(5,6)(7,8)(9,10,12,17,14,18,13,16)(11,19,21,15,20,24,22,23);(6,8)(9,11,13,22,14,20,12,21)(10,15,17,24,18,23,16,19);(11,20)(15,23)(19,24)(21,22)176 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 260336);