#8416840

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

4564 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)(3,7)(4,6)(5,8)(9,10,15,14)(11,19,22,16)(12,17,21,23)(13,24,20,18)(27,33)(31,34)(35,37)(39,40);(1,2,4,6)(3,5,8,7)(9,11,15,22)(10,16,14,19)(12,20,21,13)(17,24,23,18)(25,26)(27,34)(28,36)(29,30)(31,33)(32,38)(35,39)(37,40);(1,2)(3,5)(4,6)(7,8)(10,14)(16,19)(17,23)(18,24)(26,30)(31,34)(32,36)(39,40);(1,3)(2,5)(4,8)(6,7)(9,12)(10,17)(11,20)(13,22)(14,23)(15,21)(16,24)(18,19)(25,27,29,33)(26,31,30,34)(28,37,38,35)(32,40,36,39);(9,13,15,20)(10,18,14,24)(11,21,22,12)(16,23,19,17)(25,28,29,38)(26,32,30,36)(27,35,33,37)(31,39,34,40);(9,14,15,10)(11,19,22,16)(12,23,21,17)(13,24,20,18)(25,26)(27,31)(28,32)(29,30)(33,34)(35,39)(36,38)(37,40)

Polycyclic presentation:

2^9|a^2=h,b^2=g,d^2=ghi,e^2=gi,f^2=h,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dghi,e^b=egi,f^c=fgi,e^d=egi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8324036);