#9206536

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

4293 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)(9,10,14,15)(11,17,18,12)(13,21,23,16)(19,20,24,22)(25,26,31,36)(27,34,29,32)(28,38,37,33)(30,35,39,40);(1,3)(2,6)(4,8)(5,7)(9,10,14,15)(11,12,18,17)(13,16,23,21)(19,20,24,22)(25,27,31,29)(26,32,36,34)(28,39,37,30)(33,40,38,35);(1,4)(2,7)(3,8)(5,6)(10,15)(11,18)(16,21)(19,24)(25,28)(26,33)(27,30)(29,39)(31,37)(32,35)(34,40)(36,38);(1,4,5,6)(2,3,8,7)(9,11,14,18)(10,12,15,17)(13,19,23,24)(16,20,21,22)(25,29,31,27)(26,34,36,32)(28,39,37,30)(33,40,38,35);(1,4)(2,3)(5,6)(7,8)(9,12)(10,11)(13,22)(14,17)(15,18)(16,24)(19,21)(20,23)(25,30,31,39)(26,35,36,40)(27,37,29,28)(32,38,34,33);(1,3)(2,6)(4,8)(5,7)(9,13,14,23)(10,16,15,21)(11,19,18,24)(12,20,17,22)(25,26)(27,32)(28,33)(29,34)(30,35)(31,36)(37,38)(39,40)

Polycyclic presentation:

2^9|a^2=i,b^2=i,d^2=h,e^2=ghi,f^2=gh,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, 9113732);