#9075936

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

4373 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)(11,19)(12,17)(13,20)(14,15)(16,22)(18,24)(21,23)(25,26,31,35)(27,32,36,40)(28,30,34,39)(29,33,37,38);(1,3,6,5)(2,4,7,8)(10,15)(12,23)(16,19)(18,20)(25,27)(26,32)(28,37)(29,34)(30,38)(31,36)(33,39)(35,40);(1,4,6,8)(2,5,7,3)(10,15)(12,23)(16,19)(18,20)(27,36)(29,37)(32,40)(33,38);(1,4,6,8)(2,3,7,5)(9,11)(10,16)(12,18)(13,17)(14,22)(15,19)(20,23)(21,24)(25,28)(26,30)(27,37)(29,36)(31,34)(32,38)(33,40)(35,39);(1,4)(2,3)(5,7)(6,8)(9,12,14,23)(10,17,15,21)(11,20,22,18)(13,16,24,19)(25,29,31,37)(26,33,35,38)(27,34,36,28)(30,32,39,40);(1,5,6,3)(2,4,7,8)(9,13,14,24)(10,18,15,20)(11,21,22,17)(12,16,23,19)(25,30)(26,34)(27,38)(28,35)(29,32)(31,39)(33,36)(37,40)

Polycyclic presentation:

2^9|a^2=ghi,b^2=gh,c^2=gh,d^2=gh,e^2=hi,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,e^c=egi,e^d=egi,f^d=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, 8983132);