Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #360 |
| automorphism | ? |
| homology | Abelianization (H1) = C4×C16 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C16 | #102 |
| commutator | C8 | C4×C16 |
| Frattini, cosocle | C2×C4×C16 | C22 |
| socle | C2 | #7151 |
| prime radical | C23 | #364 |
| 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 | |
145 conjugacy classes of subgroups; only the 84 normal subgroups are shown.
This group has 145 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:
(1,2,5,12,8,11,30,40,7,15,29,44,27,41,55,63,9,16,31,45,6,32,28,56,25,14,46,43,4,13,26,42)(3,17,19,47,22,36,52,59,21,35,51,58,18,10,33,34,23,37,53,60,20,38,50,61,49,48,57,64,24,39,54,62);(1,3,6,20,7,21,4,24,9,23,8,22,25,49,27,18)(2,10,13,35,14,36,32,17,16,39,41,48,15,38,11,37)(5,19,28,50,29,51,26,54,31,53,30,52,46,57,55,33)(12,34,42,58,43,59,56,47,45,62,63,64,44,61,40,60)Polycyclic presentation:
2^9|a^2=d,b^2=e,c^2=fi,d^2=g,e^2=f,f^2=i,g^2=h,h^2=i,b^a=bc,c^a=cf,e^a=ef,f^a=fi,c^b=ci128 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1216);