#8057941

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

4173 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=h,c^2=gh,d^2=gh,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, 7965137);