#503439

order	512
nilpotency class	4
isoclinism	isoclinic class of #501244
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#447
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#61091	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2931 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

940 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=i,b^2=e,c^2=efh,d^2=e,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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 410635);