#8159082

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) = 128

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=i,c^2=gi,e^2=h,f^2=gh,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, 8066278);