#8914418

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	#35405	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4266 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,d^2=g,e^2=ghi,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,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, 8821614);