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 | |
294 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 40 points:
(1,2,5,6)(3,4,7,8)(9,10,14,20,15,19,31,36,16,21,24,34,13,28,30,40)(11,12,18,26,22,17,32,33,23,27,35,37,29,25,38,39);(1,3,4,6,5,7,8,2)(9,11,15,22,16,23,13,29)(10,17,19,27,21,25,28,12)(14,18,31,32,24,35,30,38)(20,33,36,37,34,39,40,26);(9,12,24,37,15,17,30,39,16,27,14,26,13,25,31,33)(10,18,34,29,19,32,40,11,21,35,20,22,28,38,36,23)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=ci,d^b=di128 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 35877);