#9668266

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	#52353	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4097 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)(3,7)(4,6)(5,8)(9,10)(11,20)(12,13)(14,24)(15,17)(16,21)(18,23)(19,22)(25,26,31,28)(27,32,37,35)(29,33,36,38)(30,34,40,39);(1,2,4,6)(3,5,8,7)(9,11,15,21)(10,16,17,20)(12,22,23,14)(13,24,18,19)(25,27,31,37)(26,32,28,35)(29,30,36,40)(33,34,38,39);(1,2)(3,5)(4,6)(7,8)(10,17)(11,21)(12,23)(19,24)(25,26,31,28)(27,35,37,32)(29,38,36,33)(30,34,40,39);(1,3,4,8)(2,5,6,7)(9,12)(10,18)(11,22)(13,17)(14,21)(15,23)(16,19)(20,24)(25,28,31,26)(27,35,37,32)(29,33,36,38)(30,34,40,39);(9,13,15,18)(10,12,17,23)(11,19,21,24)(14,20,22,16)(25,29,31,36)(26,33,28,38)(27,30,37,40)(32,34,35,39);(9,14,15,22)(10,19,17,24)(11,12,21,23)(13,20,18,16)(25,30)(26,34)(27,36)(28,39)(29,37)(31,40)(32,38)(33,35)

Polycyclic presentation:

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

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9575462);