#8975338

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

4592 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)(3,8)(4,7)(5,6)(10,15)(11,16)(17,22)(18,23)(25,26,31,29)(27,28,35,32)(30,37,38,33)(34,39,40,36);(3,4)(7,8)(9,10,14,15)(11,12,16,19)(13,17,21,22)(18,20,23,24)(25,27,31,35)(26,32,29,28)(30,34,38,40)(33,39,37,36);(1,3)(2,7)(4,6)(5,8)(11,16)(12,19)(13,21)(17,22)(26,29)(28,32)(30,38)(34,40);(3,4)(7,8)(9,11)(10,12)(13,18)(14,16)(15,19)(17,20)(21,23)(22,24)(25,28)(26,27)(29,35)(30,36)(31,32)(33,34)(37,40)(38,39);(1,4,6,3)(2,7,5,8)(9,12)(10,16)(11,15)(13,20)(14,19)(17,23)(18,22)(21,24)(25,29,31,26)(27,32,35,28)(30,37,38,33)(34,39,40,36);(1,5,6,2)(3,7,4,8)(9,13,14,21)(10,17,15,22)(11,18,16,23)(12,20,19,24)(25,30)(26,33)(27,34)(28,36)(29,37)(31,38)(32,39)(35,40)

Polycyclic presentation:

2^9|a^2=hi,b^2=g,e^2=h,f^2=gh,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, 8882534);