#9062147

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

4449 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8969343);