#8424888

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

4308 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:

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

Polycyclic presentation:

2^9|a^2=g,c^2=ghi,d^2=h,e^2=g,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,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, 8332084);