Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #151342 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #12187 |
| commutator, Frattini, cosocle | #379 | C23 |
| prime radical | #12090 | C2 |
| LCS | G ▷ #379 ▷ C24 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23 ◁ C24 ◁ #379 ◁ G | |
| LECS | G ▷ #379 ▷ C24 ▷ C22 ▷ C2 ▷ C1 | |
504 conjugacy classes of subgroups; only the 36 normal subgroups are shown.
This group has 504 conjugacy classes of subgroups — too many to draw. Only its 36 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:
(2,6)(7,14)(8,13)(9,16)(12,15);(1,2)(3,7)(4,6)(5,8)(9,12)(10,15)(11,13)(14,16);(1,3)(2,7)(4,9,11,16)(5,10)(6,12,13,14)(8,15)Polycyclic presentation:
2^9|c^2=fgh,d^2=f,b^a=bd,c^a=ce,d^a=df,g^a=gh,d^b=df,e^b=egh,g^b=gi,h^b=hi,d^c=dg,e^c=eh,f^c=fi,g^c=gi,e^d=ehi,g^d=gi32 conjugacy classes — view the full character table.
| maximal subgroup of | #72216457, #72237728, #72237742, … |
| maximal normal subgroup of | #72216457, #72237728, #72237742, … |
| maximal quotient of | #72238057, #72238058, #72238059, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 58538);