#8621916

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	#46725	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4311 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=gi,d^2=g,e^2=gi,f^2=h,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, 8529112);