Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #511839 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #3027 |
| commutator, Frattini, cosocle | C2×#41 | C24 |
| socle | C2 | #32899 |
| LCS | G ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C2×C4 ◁ C4×#41 ◁ G | |
| LECS | G ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1 | |
2415 conjugacy classes of subgroups; only the 184 normal subgroups are shown.
This group has 2415 conjugacy classes of subgroups — too many to draw. Only its 184 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 — 4 generators on 16 points:
(1,2)(3,8)(4,7)(5,13)(6,9)(10,15)(11,14)(12,16);(2,7)(8,15)(9,14)(13,16);(4,11)(10,12);(1,3,6,5)(4,10,11,12)Polycyclic presentation:
2^9|d^2=h,g^2=i,b^a=be,c^a=cf,d^a=dg,g^a=gi,h^a=hi,f^b=fhi,e^c=eh,f^e=fi65 conjugacy classes — view the full character table.
| maximal subgroup of | #72231252, #72231703, #72234102, … |
| maximal normal subgroup of | #72231252, #72231703, #72234102, … |
| maximal quotient of | #72234084, #72257242, #72260934, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 419035);