#8339839

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	#52249	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4210 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=ghi,c^2=gi,d^2=hi,e^2=h,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,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, 8247035);