#8659367

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

4659 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=gh,c^2=g,e^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, 8566563);