Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #1426 |
| automorphism | ? |
| homology | Abelianization (H1) = C4×C8 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C8 | #367 |
| commutator | C16 | C4×C8 |
| Frattini, cosocle | C2×C4×C16 | C22 |
| socle | C2 | #7171 |
| prime radical | C23 | #359 |
| LCS | G ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C8 ◁ C22×C8 ◁ C2×C4×C8 ◁ C2×C4×C16 ◁ G | |
| LECS | G ▷ C2×C4×C16 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
153 conjugacy classes of subgroups; only the 80 normal subgroups are shown.
This group has 153 conjugacy classes of subgroups — too many to draw. Only its 80 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:
(1,2,5,12,8,14,7,33,9,16,29,13,34,15,35,41)(3,17,19,48,22,49,57,63,23,51,28,62,60,50,21,59)(4,24,25,44,6,30,26,11,27,42,31,43,32,61,55,45)(10,36,37,58,38,53,47,18,40,52,46,20,39,56,64,54);(1,3,6,20,29,19,55,56,8,22,27,54,35,57,25,36,9,23,32,58,5,28,26,53,34,60,4,18,7,21,31,52)(2,10,11,17,41,47,30,63,15,39,44,50,13,37,24,48,16,40,45,51,33,64,61,59,14,38,43,49,12,46,42,62)Polycyclic presentation:
2^9|a^2=d,b^2=e,c^2=fhi,d^2=g,e^2=fg,f^2=hi,g^2=i,h^2=i,b^a=bc,c^a=cf,e^a=efhi,f^a=fh,h^a=hi,d^b=di104 conjugacy classes — view the full character table.
| maximal subgroup of | #72225304, #72225312, #72225313, … |
| maximal normal subgroup of | #72225304, #72225312, #72225313, … |
| maximal quotient of | #72227005 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1249);