#9669366

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

4308 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=g,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,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, 9576562);