#8069826

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	C₂×#560	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4425 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=g,c^2=h,b^a=bg,c^a=ch,e^a=ei,c^b=ci,e^c=ei,f^c=fh,e^d=ehi,f^d=fi

Conjugacy classes & character table

95 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7977022);