#8119336

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

4387 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,c^2=hi,d^2=ghi,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,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, 8026532);