#8256585

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

4414 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=i,c^2=g,d^2=h,e^2=ghi,f^2=hi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,f^c=fgi

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, 8163781);