#8402272

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

4467 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)(25,26)(27,32)(28,36)(29,34)(30,31)(33,38)(35,40)(37,39);(1,2,6,5)(3,8,4,7)(9,10)(11,18)(12,13)(14,15)(16,21)(17,23)(19,20)(22,24)(25,26)(27,32)(28,36)(29,34)(30,31)(33,38)(35,40)(37,39);(1,3,6,4)(2,7,5,8)(10,15)(13,17)(16,18)(20,24)(26,31)(29,39)(32,40)(33,36);(1,4,6,3)(2,8,5,7)(9,11,14,21)(10,16,15,18)(12,22,23,19)(13,20,17,24)(25,27)(26,32)(28,37)(29,33)(30,35)(31,40)(34,38)(36,39);(1,5,6,2)(3,8,4,7)(9,12)(10,17)(11,19)(13,15)(14,23)(16,20)(18,24)(21,22)(25,28,30,38)(26,33,31,36)(27,34,35,37)(29,32,39,40);(9,13,14,17)(10,12,15,23)(11,20,21,24)(16,22,18,19)(25,29,30,39)(26,34,31,37)(27,33,35,36)(28,32,38,40)

Polycyclic presentation:

2^9|b^2=g,c^2=g,d^2=ghi,e^2=h,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,e^b=egi,f^c=fgi,e^d=egi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8309468);