#9683066

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

4271 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=g,c^2=gi,d^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,f^e=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9590262);