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) = C22×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 | #532 | C2×C4 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C8 ◁ C2×C4×C16 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
338 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 40 points:
(2,6)(3,4)(7,8)(9,10,14,20,15,19,31,36,16,21,32,37,13,28,30,40)(11,22,24,38,23,17,33,34,25,12,18,26,29,27,35,39);(1,2,4,8,5,7,3,6)(9,11,15,23,16,25,13,29)(10,17,19,12,21,27,28,22)(14,24,31,33,32,18,30,35)(20,34,36,26,37,39,40,38);(2,7)(6,8)(9,12,14,26,15,27,31,39,16,22,32,38,13,17,30,34)(10,18,20,29,19,35,36,11,21,24,37,23,28,33,40,25)Polycyclic presentation:
2^9|a^2=e,b^2=f,c^2=e,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, 35883);