#8231348

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

4416 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=g,c^2=h,d^2=gh,e^2=g,f^2=i,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,f^c=fgi

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, 8138544);