#8125468

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

4369 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=i,c^2=g,d^2=i,e^2=h,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,e^c=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, 8032664);