#9009272

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	#59755	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4294 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)(3,8)(4,7)(5,6)(25,26)(27,34)(28,31)(29,32)(30,33)(35,37)(36,39)(38,40);(3,4)(7,8)(9,10)(11,15)(12,16)(13,14)(17,21)(18,23)(19,22)(20,24)(25,27)(26,28)(29,35)(30,31)(32,36)(33,34)(37,40)(38,39);(1,3)(2,7)(4,6)(5,8)(10,14)(11,17)(12,19)(23,24)(25,28,30,34)(26,31,33,27)(29,37,38,36)(32,35,40,39);(3,4)(7,8)(9,11,13,17)(10,15,14,21)(12,20,19,18)(16,24,22,23)(25,28)(26,27)(29,37)(30,34)(31,33)(32,39)(35,40)(36,38);(1,4,6,3)(2,7,5,8)(9,12,13,19)(10,16,14,22)(11,18,17,20)(15,23,21,24)(25,29,30,38)(26,32,33,40)(27,35,31,39)(28,36,34,37);(1,5,6,2)(3,7,4,8)(9,10)(11,15)(12,16)(13,14)(17,21)(18,23)(19,22)(20,24)(25,26)(27,28)(29,32)(30,33)(31,34)(35,36)(37,39)(38,40)

Polycyclic presentation:

2^9|c^2=g,d^2=hi,e^2=gi,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^c=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, 8916468);