#8127545

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

4447 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)(11,15)(12,19)(13,21)(14,17)(16,22)(18,20)(23,24)(26,30)(31,36)(32,37)(38,40);(1,2,4,6)(3,5,8,7)(9,11,14,19)(10,15,17,12)(13,18,22,24)(16,23,21,20)(27,33)(31,36)(34,39)(38,40);(1,2)(3,5)(4,6)(7,8)(9,10,14,17)(11,12,19,15)(13,16,22,21)(18,20,24,23)(25,26)(27,31)(28,32)(29,30)(33,36)(34,38)(35,37)(39,40);(1,3,4,8)(2,5,6,7)(9,10)(11,12)(13,16)(14,17)(15,19)(18,20)(21,22)(23,24)(25,27)(26,31)(28,34)(29,33)(30,36)(32,38)(35,39)(37,40);(9,12)(10,11)(13,20)(14,15)(16,18)(17,19)(21,24)(22,23)(25,26)(27,31)(28,32)(29,30)(33,36)(34,38)(35,37)(39,40);(9,13,14,22)(10,16,17,21)(11,18,19,24)(12,20,15,23)(25,28,29,35)(26,32,30,37)(27,34,33,39)(31,38,36,40)

Polycyclic presentation:

2^9|b^2=i,c^2=gi,d^2=g,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^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, 8034741);