#8417261

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

4395 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|b^2=gi,c^2=h,d^2=gi,e^2=gi,f^2=h,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dghi,e^b=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, 8324457);