Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #490777 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#459 |
| commutator, Frattini, cosocle | C4×C8 | C24 |
| prime radical | C22×D8 | D4 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ C8×M16 ◁ G | |
| LECS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
504 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(2,5)(3,7)(6,8)(9,10,13,17)(11,14,20,23)(12,21,19,16)(15,24,22,18);(1,2,3,5,4,6,7,8);(10,14)(11,18)(12,19)(15,16)(17,23)(20,24)(21,22);(2,6)(5,8)Polycyclic presentation:
2^9|a^2=i,b^2=e,e^2=g,f^2=hi,h^2=i,b^a=be,c^a=cf,e^a=eg,f^a=fh,h^a=hi,d^b=dg,f^c=fh,h^c=hi65 conjugacy classes — view the full character table.
| maximal subgroup of | #72342595, SD32×AGL(1,ℤ/8), #72342608 |
| maximal normal subgroup of | #72342595, SD32×AGL(1,ℤ/8), #72342608 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 397974);