Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #220264 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | D4×C22⋊C4 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#3265 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
1362 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)(3,8)(6,7)(9,10)(11,15,16,13)(12,14);(1,2,4,6)(3,7,8,5);(2,6)(5,7)(9,11)(10,13)(12,16)(14,15);(2,7)(3,8)(5,6)Polycyclic presentation:
2^9|a^2=g,b^2=h,e^2=h,b^a=be,c^a=cf,e^a=eh,f^a=fi,c^b=ch,d^b=de,e^b=eh,g^c=gi,e^d=eh65 conjugacy classes — view the full character table.
| maximal subgroup of | #99×AGL(1,ℤ/8), #72313069, #72313082, … |
| maximal normal subgroup of | #99×AGL(1,ℤ/8), #72313069, #72313082, … |
| maximal quotient of | #72313509, #72313663 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 127461);