#9017417

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	#2818	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4221 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)(25,26,31,29)(27,35,36,32)(28,34,30,33)(37,39,38,40);(3,8)(6,7)(9,10,15,19)(11,20,22,16)(12,14,18,24)(13,17,21,23)(26,29)(27,36)(33,34)(37,38);(1,3,5,8)(2,6,4,7)(9,11,15,22)(10,16,19,20)(12,13,18,21)(14,23,24,17)(25,27,31,36)(26,32,29,35)(28,37,30,38)(33,39,34,40);(3,8)(6,7)(9,12,15,18)(10,14,19,24)(11,21,22,13)(16,17,20,23)(25,28,31,30)(26,33,29,34)(27,37,36,38)(32,39,35,40);(1,3)(2,7)(4,6)(5,8)(9,13,15,21)(10,17,19,23)(11,12,22,18)(14,16,24,20)(25,29,31,26)(27,35,36,32)(28,34,30,33)(37,40,38,39);(1,4,5,2)(3,6,8,7)(9,14)(10,18)(11,17)(12,19)(13,16)(15,24)(20,21)(22,23)(25,30,31,28)(26,34,29,33)(27,38,36,37)(32,40,35,39)

Polycyclic presentation:

2^9|a^2=g,b^2=hi,c^2=i,d^2=ghi,e^2=ghi,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^c=egi,f^c=fgi,e^d=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8924613);