#8159119

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

4280 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)(3,7)(4,6)(5,8)(9,10)(11,14)(12,17)(13,18)(15,20)(16,21)(19,22)(23,24)(26,29)(27,28)(34,37)(35,36);(1,2,4,6)(3,5,8,7)(10,11)(15,17)(16,18)(22,23)(25,26,31,29)(27,33,28,32)(30,34,38,37)(35,40,36,39);(1,2)(3,5)(4,6)(7,8)(10,11)(15,17)(16,18)(22,23)(25,27,31,28)(26,32,29,33)(30,35,38,36)(34,39,37,40);(1,3,4,8)(2,5,6,7)(9,11,14,10)(12,15,20,17)(13,18,21,16)(19,22,24,23)(25,28,31,27)(26,33,29,32)(30,36,38,35)(34,40,37,39);(9,12,14,20)(10,15,11,17)(13,19,21,24)(16,22,18,23)(25,29,31,26)(27,33,28,32)(30,37,38,34)(35,40,36,39);(9,13,14,21)(10,16,11,18)(12,19,20,24)(15,22,17,23)(25,30,31,38)(26,34,29,37)(27,35,28,36)(32,39,33,40)

Polycyclic presentation:

2^9|b^2=i,c^2=ghi,d^2=g,e^2=h,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi,e^d=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, 8066315);