#8726567

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

4313 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:

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

Polycyclic presentation:

2^9|b^2=ghi,c^2=gi,d^2=h,e^2=i,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,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, 8633763);