#8743414

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	#52383	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4167 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=ghi,c^2=i,d^2=gh,e^2=gh,f^2=gh,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8650610);