#8347439

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

4700 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=g,d^2=g,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,f^c=fgi,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, 8254635);