Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #19162 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#121 |
| commutator | C42 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| socle | C22 | C2×#389 |
| prime radical | C24 | C4⋊Q8 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
630 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)(9,10,13,16,14,17,12,21)(11,18,19,15,20,23,22,24);(9,11,13,19,14,20,12,22)(10,15,16,23,17,24,21,18);(1,2,3,5,4,6,7,8);(2,6)(5,8)Polycyclic presentation:
2^9|a^2=g,b^2=g,c^2=f,e^2=h,f^2=i,g^2=h,b^a=be,c^a=cf,e^a=eh,f^a=fi,e^b=eh,d^c=di86 conjugacy classes — view the full character table.
| maximal subgroup of | #108×AGL(1,ℤ/8), #72342644, #72342647 |
| maximal normal subgroup of | #108×AGL(1,ℤ/8), #72342644, #72342647 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 111126);