#8759335

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	#47826	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4535 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|b^2=gh,c^2=g,d^2=h,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^c=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, 8666531);