#9017354

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

4423 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)(3,4)(5,6)(7,8)(10,15)(11,18)(17,22)(19,24)(25,26,31,35)(27,32,30,40)(28,29,34,33)(36,38,37,39);(2,6)(3,7)(9,10,14,15)(11,16,18,12)(13,17,21,22)(19,23,24,20)(25,27,31,30)(26,32,35,40)(28,37,34,36)(29,39,33,38);(1,3,5,7)(2,4,6,8)(11,18)(12,16)(13,21)(17,22)(25,28,31,34)(26,33,35,29)(27,36,30,37)(32,39,40,38);(1,4)(2,7)(3,6)(5,8)(9,11,14,18)(10,16,15,12)(13,19,21,24)(17,23,22,20)(25,28)(26,29)(27,37)(30,36)(31,34)(32,39)(33,35)(38,40);(9,12,14,16)(10,11,15,18)(13,20,21,23)(17,19,22,24)(25,29)(26,34)(27,38)(28,35)(30,39)(31,33)(32,37)(36,40);(9,13)(10,17)(11,19)(12,20)(14,21)(15,22)(16,23)(18,24)(25,30,31,27)(26,32,35,40)(28,36,34,37)(29,39,33,38)

Polycyclic presentation:

2^9|a^2=h,b^2=gi,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, 8924550);