nLab image of J -- table

	$n$	0	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16
Whitehead tower of orthogonal group		orientation	spin group		string group				fivebrane group	2-orient group		2-spin group	ninebrane group
higher versions		special orthogonal group	spin group		string 2-group				fivebrane 6-group				ninebrane 10-group
homotopy groups of stable orthogonal group	$\pi_n(O)$	$\mathbb{Z}_2$	$\mathbb{Z}_2$	0	$\mathbb{Z}$	0	0	0	$\mathbb{Z}$	$\mathbb{Z}_2$	$\mathbb{Z}_2$	0	$\mathbb{Z}$	0	0	0	$\mathbb{Z}$	$\mathbb{Z}_2$
stable homotopy groups of spheres	$\pi_n(\mathbb{S})$	$\mathbb{Z}$	$\mathbb{Z}_2$	$\mathbb{Z}_2$	$\mathbb{Z}_{24}$	0	0	$\mathbb{Z}_2$	$\mathbb{Z}_{240}$	$\mathbb{Z}_2 \oplus \mathbb{Z}_2$	$\mathbb{Z}_2 \oplus \mathbb{Z}_2 \oplus \mathbb{Z}_2$	$\mathbb{Z}_6$	$\mathbb{Z}_{504}$	0	$\mathbb{Z}_3$	$\mathbb{Z}_2 \oplus \mathbb{Z}_2$	$\mathbb{Z}_{480} \oplus \mathbb{Z}_2$	$\mathbb{Z}_2 \oplus \mathbb{Z}_2$
image of J-homomorphism	$im(\pi_n(J))$	0	$\mathbb{Z}_2$	0	$\mathbb{Z}_{24}$	0	0	0	$\mathbb{Z}_{240}$	$\mathbb{Z}_2$	$\mathbb{Z}_2$	0	$\mathbb{Z}_{504}$	0	0	0	$\mathbb{Z}_{480}$	$\mathbb{Z}_2$