#8806894

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

4306 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)(4,5)(6,8)(9,10,14,15)(11,17,19,12)(13,21,23,16)(18,20,22,24)(27,30)(32,34)(35,37)(39,40);(2,5)(6,7)(9,11,14,19)(10,12,15,17)(13,22,23,18)(16,24,21,20)(25,26,31,28)(27,32,30,34)(29,36,38,33)(35,40,37,39);(1,2)(3,6)(4,5)(7,8)(10,15)(12,17)(16,21)(20,24)(25,27,31,30)(26,32,28,34)(29,35,38,37)(33,39,36,40);(1,3)(2,6)(4,8)(5,7)(9,10)(11,12)(13,16)(14,15)(17,19)(18,20)(21,23)(22,24)(25,28,31,26)(27,34,30,32)(29,36,38,33)(35,40,37,39);(9,12)(10,11)(13,20)(14,17)(15,19)(16,18)(21,22)(23,24)(25,29,31,38)(26,33,28,36)(27,35,30,37)(32,39,34,40);(9,13,14,23)(10,16,15,21)(11,18,19,22)(12,20,17,24)(25,30,31,27)(26,34,28,32)(29,37,38,35)(33,40,36,39)

Polycyclic presentation:

2^9|a^2=gi,b^2=h,c^2=ghi,d^2=ghi,e^2=ghi,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,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, 8714090);