Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #19162 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
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×#380 |
| prime radical | C22×#41 | D4 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
1059 conjugacy classes of subgroups.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 16 points:
(2,5)(3,7)(6,8)(9,10,12,15,13,11,16,14);(1,2,3,5,4,6,7,8);(9,11)(10,13)(12,14)(15,16);(2,6)(5,8)Polycyclic presentation:
2^9|a^2=g,b^2=e,e^2=h,f^2=i,g^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fi,d^b=dh,f^c=fi86 conjugacy classes — view the full character table.
| maximal subgroup of | #104×AGL(1,ℤ/8), #72342644, #72342648 |
| maximal normal subgroup of | #104×AGL(1,ℤ/8), #72342644, #72342648 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 96427);