#8759289

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

4629 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:

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

Polycyclic presentation:

2^9|a^2=h,d^2=gh,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^c=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, 8666485);