#8389283

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

4558 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)(3,8)(4,6)(5,7)(9,10)(11,14)(12,15)(13,16)(17,19)(18,21)(20,22)(23,24)(25,26,31,34)(27,32,35,30)(28,29,33,40)(36,37,38,39);(1,3,6,5)(2,7,4,8)(10,11)(12,19)(16,18)(20,24)(25,27,31,35)(26,32,34,30)(28,38,33,36)(29,39,40,37);(1,2,6,4)(3,7,5,8)(13,21)(16,18)(20,24)(22,23)(27,35)(30,32)(36,38)(37,39);(1,3,6,5)(2,7,4,8)(9,11,14,10)(12,15,19,17)(13,18,21,16)(20,22,24,23)(25,28)(26,29)(27,36)(30,39)(31,33)(32,37)(34,40)(35,38);(1,4,6,2)(3,7,5,8)(9,12,14,19)(10,15,11,17)(13,20,21,24)(16,22,18,23)(25,29)(26,33)(27,37)(28,34)(30,36)(31,40)(32,38)(35,39);(1,5,6,3)(2,7,4,8)(9,13,14,21)(10,16,11,18)(12,20,19,24)(15,22,17,23)(25,30)(26,27)(28,39)(29,36)(31,32)(33,37)(34,35)(38,40)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8296479);