#9064220

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

4402 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8971416);