#8057867

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
prime radical	#2535	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4126 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=h,d^2=h,e^2=gi,f^2=ghi,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,d^c=dh,f^e=fi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7965063);