Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33830 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C22×D8 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C22×C8 | C24 |
| socle | C22 | C22×#107 |
| prime radical | C23 | C22×Q16 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C4×Q8 ◁ C2×C8×Q8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
446 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 40 points:
(2,6)(7,8)(9,10,13,17,14,19,26,18)(11,20,24,35,21,34,39,36)(12,25,22,16,27,32,37,33)(15,28,30,38,29,40,31,23);(1,2,5,6)(3,7,4,8)(11,21)(15,29)(20,34)(23,38)(24,39)(28,40)(30,31)(35,36);(1,3,5,4)(2,7,6,8)(9,11,22,28,13,24,27,38,14,21,37,40,26,39,12,23)(10,15,25,36,18,31,33,34,19,29,32,35,17,30,16,20);(1,4,5,3)(2,7,6,8)Polycyclic presentation:
2^9|a^2=g,b^2=e,c^2=efh,d^2=e,f^2=hi,g^2=i,h^2=i,b^a=be,c^a=cf,f^a=fh,h^a=hi,c^b=ci,d^b=de86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 405157);