Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33830 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
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 | C2×C4×D8 |
| prime radical | #33604 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C22⋊C8 ◁ C8×C22⋊C4 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
618 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2)(3,4)(5,6,11,17,12,18,9,16,10,14,27,33,20,8,15,28)(7,19,24,13,25,32,29,26,23,21,34,31,30,36,22,35);(5,7,20,25,10,23,12,30)(6,13,8,26,14,31,18,35)(9,24,11,29,15,34,27,22)(16,32,17,21,28,36,33,19);(2,4)(6,14)(8,18)(13,31)(16,28)(17,33)(19,21)(26,35)(32,36);(5,8,27,16,12,6,15,33,10,18,11,28,20,14,9,17)(7,21,24,31,25,36,29,35,23,19,34,13,30,32,22,26)Polycyclic presentation:
2^9|a^2=g,b^2=egh,d^2=eh,e^2=hi,g^2=h,h^2=i,b^a=be,c^a=cf,d^b=de,e^b=eh,g^b=gh,h^b=hi,d^c=di86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 405541);