Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #501810 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#481 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| prime radical | C22×D8 | C23 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ C2×#446 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
451 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2,4,5)(3,7,8,6)(11,18)(14,19)(16,24)(20,21);(2,5)(6,7)(9,10,12,22,13,15,23,17)(11,19,21,16,18,14,20,24);(10,14)(11,20)(12,23)(15,19)(16,22)(17,24)(18,21);(1,3,4,8)(2,6,5,7)Polycyclic presentation:
2^9|a^2=e,b^2=fh,d^2=e,g^2=h,h^2=i,b^a=be,c^a=cf,d^a=de,g^a=gi,c^b=cg,f^b=fi,g^b=gh,h^b=hi,g^c=ghi,h^c=hi56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 409154);