#9206580

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	#53513	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4158 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)(4,7)(5,6)(11,17)(14,18)(19,23)(21,24)(25,26,31,32)(27,35,36,33)(28,39,40,29)(30,37,38,34);(1,2,6,5)(3,7,4,8)(9,10,15,12)(11,14,17,18)(13,20,22,16)(19,24,23,21)(26,32)(27,36)(29,39)(34,37);(1,3,6,4)(2,7,5,8)(9,11)(10,14)(12,18)(13,19)(15,17)(16,21)(20,24)(22,23)(25,27,31,36)(26,33,32,35)(28,37,40,34)(29,38,39,30);(3,4)(7,8)(9,12,15,10)(11,18,17,14)(13,20,22,16)(19,24,23,21)(25,28)(26,29)(27,37)(30,35)(31,40)(32,39)(33,38)(34,36);(1,4,6,3)(2,7,5,8)(9,13,15,22)(10,16,12,20)(11,19,17,23)(14,21,18,24)(25,29)(26,28)(27,38)(30,36)(31,39)(32,40)(33,37)(34,35);(1,5,6,2)(3,8,4,7)(9,14,15,18)(10,17,12,11)(13,21,22,24)(16,23,20,19)(25,30,31,38)(26,34,32,37)(27,29,36,39)(28,35,40,33)

Polycyclic presentation:

2^9|a^2=i,b^2=hi,c^2=g,d^2=gh,e^2=hi,f^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^d=egi,f^e=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9113776);