#8687326

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

4299 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)(11,18)(12,15)(13,20)(14,16)(17,22)(19,23)(21,24)(25,26,31,30)(27,34,29,32)(28,33,36,38)(35,40,37,39);(1,3,5,4)(2,6,8,7)(9,11,15,14)(10,16,12,18)(13,19,22,21)(17,23,20,24)(28,36)(33,38)(35,37)(39,40);(9,10,15,12)(11,18,14,16)(13,17,22,20)(19,24,21,23)(25,27,31,29)(26,32,30,34)(28,35,36,37)(33,39,38,40);(1,3,5,4)(2,7,8,6)(9,12,15,10)(11,16,14,18)(13,20,22,17)(19,23,21,24)(25,28,31,36)(26,33,30,38)(27,35,29,37)(32,39,34,40);(1,4,5,3)(2,6,8,7)(9,13)(10,17)(11,19)(12,20)(14,21)(15,22)(16,23)(18,24)(25,29,31,27)(26,34,30,32)(28,37,36,35)(33,40,38,39);(1,2)(3,6)(4,7)(5,8)(9,14,15,11)(10,16,12,18)(13,21,22,19)(17,23,20,24)(25,30,31,26)(27,34,29,32)(28,38,36,33)(35,40,37,39)

Polycyclic presentation:

2^9|a^2=hi,b^2=g,c^2=h,d^2=ghi,e^2=gh,f^2=h,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,f^d=fgi,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, 8594522);