#9789170

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	#48294	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4175 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)(11,13)(12,17)(14,23)(15,16)(18,20)(19,22)(21,24)(26,32)(28,34)(33,38)(35,37);(1,2,4,6)(3,5,8,7)(9,11)(10,13)(12,20)(14,21)(15,19)(16,22)(17,18)(23,24)(25,26,31,32)(27,33,29,38)(28,30,34,40)(35,36,37,39);(1,2)(3,5)(4,6)(7,8)(10,16)(11,19)(12,21)(18,23)(25,27,31,29)(26,33,32,38)(28,35,34,37)(30,36,40,39);(9,12,15,21)(10,17,16,24)(11,20,19,14)(13,18,22,23)(25,28,31,34)(26,30,32,40)(27,35,29,37)(33,36,38,39);(1,2)(3,7)(4,6)(5,8)(9,13)(10,11)(12,18)(14,24)(15,22)(16,19)(17,20)(21,23)(25,29,31,27)(26,33,32,38)(28,37,34,35)(30,36,40,39);(1,3)(2,5)(4,8)(6,7)(9,14,15,20)(10,18,16,23)(11,21,19,12)(13,17,22,24)(25,30)(26,34)(27,36)(28,32)(29,39)(31,40)(33,37)(35,38)

Polycyclic presentation:

2^9|b^2=ghi,c^2=g,d^2=gh,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eghi,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, 9696366);