Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1504 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C8 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C8 | C2×C22⋊C4 |
| commutator | C2×C4 | C2×C4×C8 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C22 | C8×C22⋊C4 |
| prime radical | C23 | C8×D4 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C8 ◁ C2×C4×C16 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
292 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 40 points:
(1,2)(3,4)(5,7)(6,8)(9,10,14,19,13,20,29,38,15,21,25,33,30,18,36,37)(11,12,17,27,23,26,34,40,24,28,35,39,22,16,31,32);(1,3,5,8,6,7,4,2)(9,11,13,23,15,24,30,22)(10,16,20,12,21,26,18,28)(14,17,29,34,25,35,36,31)(19,32,38,27,33,40,37,39);(1,4,6,5)(2,3,8,7)(9,12,25,39,13,26,36,32,15,28,14,27,30,16,29,40)(10,17,33,22,20,34,37,11,21,35,19,23,18,31,38,24)Polycyclic presentation:
2^9|a^2=e,b^2=f,c^2=eg,d^2=g,e^2=h,f^2=g,h^2=i,b^a=bd,d^a=dg,f^a=fgi,c^b=cgi,d^b=di128 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 35897);