#8614126

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

4415 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)(11,14)(12,17)(13,16)(15,20)(18,21)(19,23)(22,24)(25,26,32,36)(27,33,39,40)(28,30,29,34)(31,35,38,37);(1,2)(3,5)(4,6)(7,8)(10,11)(15,17)(16,18)(22,23)(25,27,32,39)(26,33,36,40)(28,37,29,35)(30,31,34,38);(2,3)(6,7)(9,10,14,11)(12,15,20,17)(13,16,21,18)(19,22,24,23)(25,28,32,29)(26,34,36,30)(27,35,39,37)(31,40,38,33);(4,8)(6,7)(9,11,14,10)(12,17,20,15)(13,18,21,16)(19,23,24,22)(25,29,32,28)(26,34,36,30)(27,37,39,35)(31,40,38,33);(1,3,5,2)(4,7,8,6)(9,12,14,20)(10,15,11,17)(13,19,21,24)(16,22,18,23)(25,30,32,34)(26,28,36,29)(27,38,39,31)(33,35,40,37);(1,4)(2,6)(3,7)(5,8)(9,13,14,21)(10,16,11,18)(12,19,20,24)(15,22,17,23)(25,31,32,38)(26,35,36,37)(27,34,39,30)(28,40,29,33)

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,c^2=h,d^2=h,e^2=i,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=egi,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, 8521322);