#8762128

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

4452 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)(3,7)(4,6)(5,8)(9,10)(11,14)(12,20)(13,16)(15,22)(17,19)(18,24)(21,23)(25,26,31,32)(27,35,36,33)(28,38,37,34)(29,30,39,40);(1,2,4,6)(3,5,8,7)(9,11,13,19)(10,14,16,17)(12,21,22,18)(15,24,20,23)(26,32)(27,36)(29,39)(34,38);(1,2)(3,5)(4,6)(7,8)(9,10,13,16)(11,17,19,14)(12,15,22,20)(18,24,21,23)(25,27,31,36)(26,33,32,35)(28,29,37,39)(30,38,40,34);(1,3,4,8)(2,5,6,7)(9,10)(11,14)(12,15)(13,16)(17,19)(18,23)(20,22)(21,24)(25,28,31,37)(26,34,32,38)(27,29,36,39)(30,35,40,33);(9,11)(10,14)(12,18)(13,19)(15,23)(16,17)(20,24)(21,22)(25,29)(26,30)(27,37)(28,36)(31,39)(32,40)(33,38)(34,35);(9,12)(10,15)(11,18)(13,22)(14,23)(16,20)(17,24)(19,21)(25,30)(26,29)(27,38)(28,35)(31,40)(32,39)(33,37)(34,36)

Polycyclic presentation:

2^9|a^2=ghi,b^2=gh,c^2=h,d^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^c=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, 8669324);