#502660

order	512
nilpotency class	4
isoclinism	isoclinic class of #501248
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#447
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#24803	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ #2929 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

776 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 40 points:

(1,2)(3,7)(4,6)(5,8)(9,10,13,19)(11,22,23,17)(12,27,29,15)(14,20,31,36)(16,30,18,21)(24,28,32,40)(25,34,39,37)(26,35,33,38);(1,3,4,8)(2,5,6,7)(9,11,13,23)(10,17,19,22)(12,28,29,40)(14,25,31,39)(15,32,27,24)(16,33,18,26)(20,37,36,34)(21,38,30,35);(9,12,27,19,16,20,14,30,13,29,15,10,18,36,31,21)(11,17,25,28,26,35,24,34,23,22,39,40,33,38,32,37);(2,6)(5,7)(10,19)(12,29)(17,22)(20,36)(21,30)(28,40)(34,37)(35,38)

Polycyclic presentation:

2^9|a^2=i,b^2=e,c^2=fhi,f^2=hi,g^2=h,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,g^a=ghi,h^a=hi,c^b=cg,f^b=fhi,g^b=ghi,h^b=hi,d^c=di

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 409856);