#9206587

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	#48784	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4171 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=ghi,c^2=g,d^2=h,e^2=ghi,f^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^d=egi,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, 9113783);