Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #152602 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #13049 |
| commutator | #127 | C22×C4 |
| Frattini, cosocle | C2×#127 | C23 |
| prime radical | #18524 | C2 |
| LCS | G ▷ #127 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23 ◁ C22×C4 ◁ C2×#127 ◁ G | |
| LECS | G ▷ C2×#127 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
650 conjugacy classes of subgroups; only the 60 normal subgroups are shown.
This group has 650 conjugacy classes of subgroups — too many to draw. Only its 60 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 16 |
A minimal faithful permutation representation — 3 generators on 16 points:
(1,2)(3,9,12,6)(4,14,15,7)(5,8)(10,13,16,11);(1,2)(3,10,13,6)(4,7)(5,8)(9,12,16,11)(14,15);(1,3)(2,6)(4,11)(5,13)(7,16)(8,10)(9,14)(12,15)Polycyclic presentation:
2^9|a^2=f,b^2=df,e^2=h,g^2=i,b^a=bd,c^a=ce,e^a=egh,g^a=gh,h^a=hi,c^b=ch,e^b=egh,h^b=hi,d^c=dg,e^c=eh,f^c=fgi,g^c=gi,e^d=eh,g^d=gi,f^e=fh,g^f=gi38 conjugacy classes — view the full character table.
| maximal subgroup of | #72216584, #72216600, #72216601, … |
| maximal normal subgroup of | #72216584, #72216600, #72216601, … |
| maximal quotient of | #72245231, #72245262, #72245414, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 59798);