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×C4 |
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×#390 |
| prime radical | #60700 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
1167 conjugacy classes of subgroups.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 16 points:
(2,7)(3,8)(4,5)(10,13,14,15);(1,2,4,7,6,8,5,3)(9,10,11,15,12,14,16,13);(1,3,5,8,6,7,4,2);(2,8)(3,7)Polycyclic presentation:
2^9|a^2=g,b^2=eg,c^2=f,e^2=hi,f^2=h,b^a=be,c^a=cf,e^a=eh,f^a=fh,d^b=dh,e^b=ei,g^b=gi,d^c=dh86 conjugacy classes — view the full character table.
| maximal subgroup of | #104×AGL(1,ℤ/8), #72342644, #72342646 |
| maximal normal subgroup of | #104×AGL(1,ℤ/8), #72342644, #72342646 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 112259);