#8898165

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4379 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=ghi,d^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,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, 8805361);