#9683963

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
prime radical	#2688	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4144 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)(11,18)(12,21)(13,15)(14,17)(16,20)(19,23)(22,24)(25,26)(27,35)(28,34)(29,33)(30,36)(31,37)(32,38)(39,40);(1,2)(3,8)(4,6)(5,7)(9,11)(10,16)(12,19)(13,18)(14,22)(15,20)(17,23)(21,24)(25,27,32,29)(26,33,38,35)(28,37,39,30)(31,34,36,40);(1,3,5,6)(2,4,7,8)(9,12,15,14)(10,17,13,21)(11,19,20,22)(16,23,18,24)(25,28,32,39)(26,34,38,40)(27,37,29,30)(31,33,36,35);(9,13,15,10)(11,18,20,16)(12,17,14,21)(19,23,22,24)(25,29,32,27)(26,35,38,33)(28,30,39,37)(31,40,36,34);(1,2)(3,4)(5,7)(6,8)(9,14,15,12)(10,17,13,21)(11,19,20,22)(16,24,18,23)(25,30)(26,36)(27,28)(29,39)(31,38)(32,37)(33,40)(34,35);(1,4)(2,6)(3,7)(5,8)(9,11)(10,16)(12,19)(13,18)(14,22)(15,20)(17,23)(21,24)(25,31,32,36)(26,37,38,30)(27,34,29,40)(28,33,39,35)

Polycyclic presentation:

2^9|b^2=gh,c^2=gi,d^2=g,e^2=h,f^2=gh,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, 9591159);