#8159411

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

4699 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)(11,17)(12,15)(13,16)(14,20)(18,22)(19,23)(21,24)(26,28)(30,38)(32,33)(35,37);(1,2,4,5)(3,6,7,8)(9,11,15,13)(10,16,12,17)(14,19,22,21)(18,23,20,24)(25,26,31,28)(27,32,29,33)(30,34,38,36)(35,39,37,40);(1,2)(3,6)(4,5)(7,8)(10,12)(16,17)(18,20)(23,24)(25,27,31,29)(26,32,28,33)(30,35,38,37)(34,39,36,40);(1,2)(3,6)(4,5)(7,8)(9,12,15,10)(11,16,13,17)(14,20,22,18)(19,23,21,24)(25,28,31,26)(27,32,29,33)(30,36,38,34)(35,39,37,40);(1,3,4,7)(2,6,5,8)(9,13,15,11)(10,17,12,16)(14,21,22,19)(18,24,20,23)(25,29,31,27)(26,33,28,32)(30,37,38,35)(34,40,36,39);(9,14)(10,18)(11,19)(12,20)(13,21)(15,22)(16,23)(17,24)(25,30)(26,34)(27,35)(28,36)(29,37)(31,38)(32,39)(33,40)

Polycyclic presentation:

2^9|b^2=g,c^2=gh,d^2=gi,e^2=g,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, 8066607);