#9057408

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4413 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)(11,17)(15,18)(19,23)(22,24)(25,26,32,37)(27,38,29,33)(28,39,40,34)(30,35,36,31);(3,7)(5,8)(9,10,13,14)(11,15,17,18)(12,16,20,21)(19,22,23,24)(25,27,32,29)(26,33,37,38)(28,36,40,30)(31,39,35,34);(1,2)(3,5)(4,6)(7,8)(10,14)(11,17)(12,20)(22,24)(25,28,32,40)(26,34,37,39)(27,36,29,30)(31,38,35,33);(9,11)(10,15)(12,19)(13,17)(14,18)(16,22)(20,23)(21,24)(25,29,32,27)(26,33,37,38)(28,36,40,30)(31,34,35,39);(1,3,4,7)(2,5,6,8)(9,10)(11,18)(12,16)(13,14)(15,17)(19,24)(20,21)(22,23)(25,30,32,36)(26,35,37,31)(27,28,29,40)(33,34,38,39);(1,2)(3,5)(4,6)(7,8)(9,12,13,20)(10,16,14,21)(11,19,17,23)(15,22,18,24)(25,31,32,35)(26,36,37,30)(27,39,29,34)(28,33,40,38)

Polycyclic presentation:

2^9|a^2=h,b^2=gi,c^2=g,d^2=g,e^2=h,f^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, 8964604);