#8700426

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	#59104	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4319 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)(4,6)(5,8)(9,10,14,18)(11,16,12,15)(13,17,22,21)(19,24,20,23)(25,26)(27,30)(28,34)(29,39)(31,33)(32,35)(36,37)(38,40);(1,2,4,6)(3,5,8,7)(9,11,14,12)(10,15,18,16)(13,19,22,20)(17,23,21,24)(25,27,32,31)(26,33,35,30)(28,36,29,38)(34,40,39,37);(1,2)(3,5)(4,6)(7,8)(9,10)(11,15)(12,16)(13,17)(14,18)(19,23)(20,24)(21,22)(25,28,32,29)(26,34,35,39)(27,36,31,38)(30,37,33,40);(1,3,4,8)(2,5,6,7)(9,10)(11,15)(12,16)(13,21)(14,18)(17,22)(19,24)(20,23)(25,29,32,28)(26,34,35,39)(27,36,31,38)(30,40,33,37);(9,12,14,11)(10,16,18,15)(13,19,22,20)(17,23,21,24)(25,30)(26,27)(28,37)(29,40)(31,35)(32,33)(34,36)(38,39);(9,13,14,22)(10,17,18,21)(11,19,12,20)(15,23,16,24)(25,31,32,27)(26,33,35,30)(28,38,29,36)(34,40,39,37)

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=ghi,d^2=gh,e^2=h,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^d=fgi,f^e=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, 8607622);