#8720413

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

4474 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)(4,5)(6,7)(9,10,14,13)(11,17,19,15)(12,20,21,16)(18,23,22,24)(25,26,31,32)(27,30,33,40)(28,29,37,38)(34,39,36,35);(1,3,6,5)(2,4,7,8)(9,11)(10,15)(12,18)(13,17)(14,19)(16,23)(20,24)(21,22)(25,27)(26,30)(28,36)(29,35)(31,33)(32,40)(34,37)(38,39);(1,3,6,5)(2,4,7,8)(12,21)(16,20)(18,22)(23,24)(26,32)(27,33)(28,37)(35,39);(2,7)(3,5)(9,12,14,21)(10,16,13,20)(11,18,19,22)(15,23,17,24)(25,28)(26,29)(27,34)(30,39)(31,37)(32,38)(33,36)(35,40);(1,4,6,8)(2,3,7,5)(9,13,14,10)(11,17,19,15)(12,20,21,16)(18,24,22,23)(25,29)(26,28)(27,35)(30,36)(31,38)(32,37)(33,39)(34,40);(1,5,6,3)(2,4,7,8)(9,11)(10,17)(12,22)(13,15)(14,19)(16,23)(18,21)(20,24)(25,30,31,40)(26,33,32,27)(28,35,37,39)(29,34,38,36)

Polycyclic presentation:

2^9|a^2=gh,b^2=hi,c^2=hi,d^2=ghi,e^2=g,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, 8627609);