Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #99 |
| automorphism | ? |
| homology | Abelianization (H1) = C4×C32 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C16 | C22⋊C4 |
| commutator | C22 | C4×C32 |
| Frattini, cosocle | C2×C4×C16 | C22 |
| socle | C22 | #1374 |
| prime radical | C23 | C4×C16 |
| LCS | G ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C16 ◁ C2×C4×C16 ◁ G | |
| LECS | G ▷ C2×C4×C16 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
142 conjugacy classes of subgroups; only the 80 normal subgroups are shown.
This group has 142 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) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 2 generators on 40 points:
(1,2,3,6,4,5,7,8)(9,10,12,18,14,20,27,35,15,11,17,22,30,24,33,38,16,21,13,19,26,34,28,36,31,25,29,23,32,37,40,39);(2,5)(3,7)(9,11,13,23,14,24,28,39,15,21,29,18,30,34,40,35,16,25,12,22,26,37,27,38,31,10,17,19,32,20,33,36)Polycyclic presentation:
2^9|a^2=d,b^2=e,d^2=g,e^2=fg,g^2=h,h^2=i,b^a=bc,c^a=cf,d^b=df176 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1360);