Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #7453 |
| automorphism | ? |
| homology | Abelianization (H1) = C42 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #1365 |
| commutator | C2×C16 | C42 |
| Frattini, cosocle | C2×C4×C16 | C22 |
| socle | C2 | #7274 |
| prime radical | C23 | #362 |
| LCS | G ▷ C2×C16 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ C2×C4×C8 ◁ C2×C4×C16 ◁ G | |
| LECS | G ▷ C2×C4×C16 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
141 conjugacy classes of subgroups; only the 76 normal subgroups are shown.
This group has 141 conjugacy classes of subgroups — too many to draw. Only its 76 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:
(1,2,5,12,9,15,8,14)(3,16,18,48,21,50,29,36)(4,22,25,58,6,30,28,59)(7,32,24,46,34,41,26,13)(10,19,35,17,39,23,38,53)(11,40,42,31,44,61,43,27)(20,56,54,49,33,57,51,64)(37,52,47,60,45,62,63,55);(1,3,6,19,24,54,61,52,8,18,25,17,7,33,31,60,9,21,4,23,26,51,40,62,5,29,28,53,34,20,27,55)(2,10,11,36,41,63,59,64,12,38,42,50,13,45,22,57,15,39,44,48,32,47,58,49,14,35,43,16,46,37,30,56)Polycyclic presentation:
2^9|a^2=d,b^2=e,c^2=fhi,d^2=i,e^2=fg,f^2=hi,g^2=i,h^2=i,b^a=bc,c^a=cf,e^a=efghi,f^a=fh,g^a=gi,h^a=hi,c^b=cg,d^b=di,f^b=fi,g^b=gi74 conjugacy classes — view the full character table.
| maximal subgroup of | #72225338, #72225341, #72225343, … |
| maximal normal subgroup of | #72225338, #72225341, #72225343, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1561);