Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #135516 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #8746 |
| commutator | C22×D4 | C22×C4 |
| Frattini, cosocle | C23×D4 | C23 |
| prime radical | #3484 | C22 |
| LCS | G ▷ C22×D4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23 ◁ C23×D4 ◁ G | |
| LECS | G ▷ C23×D4 ▷ C23 ▷ C2 ▷ C1 | |
571 conjugacy classes of subgroups; only the 60 normal subgroups are shown.
This group has 571 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:
(1,2,6,9)(3,4,11,12)(5,13)(7,15)(8,14)(10,16);(1,3)(2,7)(4,8)(5,10)(6,11)(9,16)(12,13)(14,15);(1,4,5,7)(2,3,8,10)(6,12,14,16)(9,11,13,15)Polycyclic presentation:
2^9|a^2=f,c^2=d,b^a=bd,c^a=ce,d^a=dg,e^a=eh,g^a=gi,e^b=ei,f^b=fg,h^b=hi,e^c=eg,f^c=fh,g^c=gi,h^c=hi,e^d=ei,f^d=fi38 conjugacy classes — view the full character table.
| maximal subgroup of | #72216336, #72216344, #72216376, … |
| maximal normal subgroup of | #72216336, #72216344, #72216376, … |
| maximal quotient of | #72241343, #72252378, #72258334, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 42712);