Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #365 |
| automorphism | ? |
| homology | Abelianization (H1) = C4×C16 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C16 | #106 |
| commutator | C8 | C4×C16 |
| Frattini, cosocle | C2×C4×C16 | C22 |
| socle | C2 | #7167 |
| prime radical | C23 | #350 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C16 ◁ C22×C16 ◁ C2×C4×C16 ◁ G | |
| LECS | G ▷ C2×C4×C16 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
155 conjugacy classes of subgroups; only the 84 normal subgroups are shown.
This group has 155 conjugacy classes of subgroups — too many to draw. Only its 84 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 2 generators on 64 points:
(1,2,5,13,8,12,32,42,9,15,27,46,4,25,26,58,10,16,33,47,28,44,35,14,7,36,31,43,29,59,6,34)(3,17,19,39,22,40,54,48,23,49,55,61,18,41,20,45,24,50,30,60,52,62,56,63,21,57,51,64,53,11,37,38);(1,3,6,20,29,53,31,55,7,21,35,54,28,52,33,19,10,24,26,37,4,18,27,51,9,23,32,56,8,22,5,30)(2,11,14,39,25,49,13,45,36,62,58,64,12,17,43,48,16,41,42,60,59,57,47,38,15,40,34,61,44,50,46,63)Polycyclic presentation:
2^9|a^2=d,b^2=e,c^2=f,d^2=g,e^2=fg,f^2=i,g^2=h,h^2=i,b^a=bc,c^a=cf,e^a=ef,f^a=fi,d^b=di128 conjugacy classes — view the full character table.
| maximal subgroup of | #72225363, #72225379, #72225381, … |
| maximal normal subgroup of | #72225363, #72225379, #72225381, … |
| maximal quotient of | #72227004 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1237);