#502654

order	512
nilpotency class	4
isoclinism	isoclinic class of #501244
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	#61262	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ #2929 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

798 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)(11,21)(12,26)(13,29)(14,28)(16,31)(17,18)(19,30)(22,35)(23,24)(25,38)(27,34)(32,40)(33,39)(36,37);(1,3,4,8)(2,5,6,7)(9,10)(11,22)(12,27)(13,17)(14,30)(15,20)(16,23)(18,29)(19,28)(21,35)(24,31)(25,40)(26,34)(32,38)(33,37)(36,39);(9,11,13,23,14,24,12,21,15,25,26,37,28,36,29,38)(10,16,27,40,19,32,17,31,20,33,18,22,30,35,34,39);(2,6)(5,7)

Polycyclic presentation:

2^9|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

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, 409850);