#8636281

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	#45046	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4306 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=ghi,c^2=g,d^2=h,e^2=gh,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=egi,f^d=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, 8543477);