#8383484

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4163 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=g,c^2=ghi,d^2=ghi,e^2=h,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,f^c=fgi,e^d=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8290680);