#9016637

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

4440 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)(9,10,15,19)(11,13,20,16)(12,17,22,24)(14,18,23,21);(3,4)(7,8)(9,11)(10,16)(12,14)(13,19)(15,20)(17,21)(18,24)(22,23)(25,26,31,35)(27,36,38,32)(28,30,34,40)(29,33,37,39);(1,3)(2,7)(4,6)(5,8)(9,12,15,22)(10,17,19,24)(11,14,20,23)(13,18,16,21)(25,27,31,38)(26,32,35,36)(28,29,34,37)(30,39,40,33);(3,4)(7,8)(9,13,15,16)(10,11,19,20)(12,21,22,18)(14,17,23,24)(25,28,31,34)(26,30,35,40)(27,37,38,29)(32,33,36,39);(1,4,6,3)(2,7,5,8)(9,12)(10,17)(11,14)(13,18)(15,22)(16,21)(19,24)(20,23)(25,29,31,37)(26,33,35,39)(27,28,38,34)(30,32,40,36);(1,5,6,2)(3,7,4,8)(9,14)(10,18)(11,12)(13,17)(15,23)(16,24)(19,21)(20,22)(25,30)(26,34)(27,33)(28,35)(29,32)(31,40)(36,37)(38,39)

Polycyclic presentation:

2^9|a^2=g,b^2=hi,c^2=ghi,d^2=ghi,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, 8923833);