#505447

order	512
nilpotency class	4
isoclinism	isoclinic class of #501868
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#3109
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂⁴	AGL(1,ℤ/8)
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #1753 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

404 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)(4,8)(5,6)(9,10)(11,16)(12,18)(13,17)(14,29)(15,20)(19,28)(21,32)(22,33)(23,34)(24,39)(25,35)(26,36)(27,37)(30,40)(31,38);(2,6)(7,8)(9,11,21,25,13,26,15,31)(10,16,32,35,17,36,20,38)(12,23,28,22,27,24,14,30)(18,34,19,33,37,39,29,40);(1,3,5,4)(2,7,6,8)(10,17)(11,22)(12,14)(15,21)(16,33)(18,19)(23,25)(24,31)(26,30)(27,28)(29,37)(34,35)(36,40)(38,39);(1,4,5,3)(2,7,6,8)(9,12,15,14,13,27,21,28)(10,18,20,29,17,37,32,19)(11,23,31,30,26,24,25,22)(16,34,38,40,36,39,35,33)

Polycyclic presentation:

2^9|b^2=fh,c^2=e,d^2=efhi,g^2=h,h^2=i,b^a=be,c^a=cf,d^a=de,g^a=gi,c^b=cg,f^b=fi,g^b=gh,h^b=hi,d^c=dh,g^c=ghi,h^c=hi,g^d=gi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 412643);