#9140567

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	#52031	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4701 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)(5,8)(6,7)(9,10,15,17)(11,20,22,16)(12,23,19,14)(13,18,24,21)(25,26)(27,36)(28,33)(29,30)(31,35)(32,39)(34,38)(37,40);(1,3,5,7)(2,4,8,6)(9,11,15,22)(10,16,17,20)(12,21,19,18)(13,23,24,14)(25,27)(26,32)(28,37)(29,38)(30,33)(31,39)(34,40)(35,36);(1,3)(2,6)(4,8)(5,7)(10,17)(12,19)(16,20)(18,21)(25,28,31,38)(26,33,35,34)(27,37,39,29)(30,36,40,32);(1,3,5,7)(2,4,8,6)(9,12,15,19)(10,14,17,23)(11,21,22,18)(13,20,24,16)(25,28)(26,34)(27,37)(29,39)(30,36)(31,38)(32,40)(33,35);(1,3)(2,4)(5,7)(6,8)(9,13,15,24)(10,18,17,21)(11,14,22,23)(12,16,19,20)(25,29,31,37)(26,30,35,40)(27,38,39,28)(32,33,36,34);(1,4)(2,7)(3,8)(5,6)(9,14)(10,19)(11,13)(12,17)(15,23)(16,18)(20,21)(22,24)(25,30)(26,29)(27,33)(28,36)(31,40)(32,38)(34,39)(35,37)

Polycyclic presentation:

2^9|a^2=hi,b^2=h,c^2=ghi,d^2=h,e^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^d=egi,f^e=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9047763);