#8720752

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

4460 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=i,c^2=ghi,d^2=i,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=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, 8627948);