Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1462 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C16 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C2×C22⋊C4 |
| commutator | C23 | C22×C16 |
| Frattini, cosocle | C23×C8 | C23 |
| socle | C23 | C2×C22⋊C8 |
| prime radical | C24 | C22×C8 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
522 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 3 generators on 28 points:
(2,4)(5,6)(7,11,12,9)(8,10)(13,14,15,18,16,19,21,24,17,20,22,25,23,26,27,28);(1,2,3,4);(5,7,8,12)(6,9,10,11)Polycyclic presentation:
2^9|a^2=f,b^2=d,c^2=g,f^2=h,h^2=i,b^a=bd,c^a=ce,e^a=eg,f^c=fg128 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 41442);