#9053885

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	#40552	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4176 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=gi,c^2=g,d^2=g,e^2=gi,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dgi,e^c=egi,f^c=fgi,e^d=egi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8961081);