#8159257

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

4321 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)(9,10,14,16)(11,17,12,15)(13,21,22,18)(19,23,20,24)(25,26)(27,34)(28,31)(29,33)(30,36)(32,37)(35,39)(38,40);(1,2,4,6)(3,5,8,7)(9,11,14,12)(10,15,16,17)(13,19,22,20)(18,23,21,24)(25,27)(26,30)(28,35)(29,36)(31,38)(32,39)(33,34)(37,40);(1,2)(3,5)(4,6)(7,8)(10,16)(15,17)(18,21)(23,24)(25,28,29,37)(26,31,33,32)(27,35,36,40)(30,38,34,39);(1,3)(2,5)(4,8)(6,7)(9,10)(11,17)(12,15)(13,18)(14,16)(19,24)(20,23)(21,22)(25,28)(26,32)(27,35)(29,37)(30,39)(31,33)(34,38)(36,40);(9,12,14,11)(10,17,16,15)(13,20,22,19)(18,24,21,23)(25,26)(27,30)(28,31)(29,33)(32,37)(34,36)(35,38)(39,40);(9,13,14,22)(10,18,16,21)(11,19,12,20)(15,23,17,24)(25,27)(26,30)(28,35)(29,36)(31,38)(32,39)(33,34)(37,40)

Polycyclic presentation:

2^9|a^2=h,b^2=hi,c^2=ghi,e^2=h,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi,e^d=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8066453);