#8662892

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

4472 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)(9,10,15,18)(11,19,21,16)(12,22,20,14)(13,17,23,24)(25,26)(27,34)(28,31)(29,36)(30,32)(33,38)(35,39)(37,40);(1,2,4,6)(3,5,8,7)(9,11,15,21)(10,16,18,19)(12,13,20,23)(14,17,22,24)(25,27,31,30)(26,32,28,34)(29,35,38,37)(33,39,36,40);(1,2)(3,5)(4,6)(7,8)(25,26,31,28)(27,34,30,32)(29,33,38,36)(35,40,37,39);(1,3,4,8)(2,5,6,7)(9,12,15,20)(10,14,18,22)(11,13,21,23)(16,17,19,24)(25,28,31,26)(27,32,30,34)(29,36,38,33)(35,39,37,40);(9,13)(10,17)(11,20)(12,21)(14,19)(15,23)(16,22)(18,24)(25,29)(26,33)(27,35)(28,36)(30,37)(31,38)(32,39)(34,40);(9,14)(10,12)(11,17)(13,16)(15,22)(18,20)(19,23)(21,24)(25,30,31,27)(26,32,28,34)(29,37,38,35)(33,39,36,40)

Polycyclic presentation:

2^9|a^2=gi,b^2=g,c^2=h,d^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,f^d=fgi,f^e=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, 8570088);