#8339867

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	#39118	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4292 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=i,c^2=hi,d^2=hi,e^2=h,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, 8247063);