#9010567

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

4401 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)(9,10)(11,12)(13,21)(14,16)(15,17)(18,22)(19,23)(20,24)(25,26,30,33)(27,31,34,38)(28,32,37,29)(35,39,40,36);(3,4)(7,8)(9,11,14,17)(10,15,16,12)(13,19,22,24)(18,23,21,20)(25,27)(26,31)(28,35)(29,36)(30,34)(32,39)(33,38)(37,40);(1,3)(2,7)(4,6)(5,8)(10,16)(12,15)(13,22)(19,24)(27,34)(28,37)(29,32)(31,38);(3,4)(7,8)(9,12,14,15)(10,17,16,11)(13,20,22,23)(18,24,21,19)(25,28,30,37)(26,32,33,29)(27,35,34,40)(31,39,38,36);(1,4,6,3)(2,7,5,8)(9,10)(11,12)(13,18)(14,16)(15,17)(19,20)(21,22)(23,24)(25,29,30,32)(26,28,33,37)(27,36,34,39)(31,35,38,40);(1,5,6,2)(3,7,4,8)(9,13,14,22)(10,18,16,21)(11,19,17,24)(12,20,15,23)(25,27)(26,31)(28,35)(29,36)(30,34)(32,39)(33,38)(37,40)

Polycyclic presentation:

2^9|a^2=ghi,b^2=g,d^2=hi,e^2=i,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^c=egi,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, 8917763);