#8343528

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
prime radical	#42503	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4231 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)(3,8)(4,6)(5,7)(9,10,15,19)(11,16,21,14)(12,17,13,18)(20,23,22,24)(25,26,31,33)(27,32,28,36)(29,30,39,40)(34,38,37,35);(1,2,6,4)(3,8,5,7)(9,11)(10,16)(12,22)(13,20)(14,19)(15,21)(17,24)(18,23)(25,27,31,28)(26,32,33,36)(29,37,39,34)(30,35,40,38);(1,2,6,4)(3,7,5,8)(9,12,15,13)(10,17,19,18)(11,20,21,22)(14,24,16,23)(25,28,31,27)(26,32,33,36)(29,37,39,34)(30,38,40,35);(1,3,6,5)(2,7,4,8)(9,11)(10,16)(12,20)(13,22)(14,19)(15,21)(17,23)(18,24)(25,29)(26,30)(27,34)(28,37)(31,39)(32,38)(33,40)(35,36);(1,4,6,2)(3,7,5,8)(9,13,15,12)(10,18,19,17)(11,20,21,22)(14,24,16,23)(25,30)(26,29)(27,35)(28,38)(31,40)(32,37)(33,39)(34,36);(1,5,6,3)(2,7,4,8)(9,14,15,16)(10,11,19,21)(12,23,13,24)(17,22,18,20)

Polycyclic presentation:

2^9|a^2=ghi,b^2=h,c^2=hi,d^2=g,e^2=gi,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,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, 8250724);