#8700454

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

4313 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)(3,5)(4,7)(6,8)(10,15)(16,17)(18,21)(23,24)(26,30)(27,33)(32,36)(34,35);(2,3)(6,7)(9,10,14,15)(11,16,12,17)(13,18,22,21)(19,23,20,24)(25,26)(27,31)(28,32)(29,30)(33,38)(34,39)(35,40)(36,37);(1,2,5,3)(4,6,8,7)(9,11,14,12)(10,16,15,17)(13,19,22,20)(18,23,21,24)(25,26)(27,31)(28,32)(29,30)(33,38)(34,39)(35,40)(36,37);(1,3,5,2)(4,7,8,6)(9,10)(11,16)(12,17)(13,21)(14,15)(18,22)(19,24)(20,23)(25,27,29,33)(26,31,30,38)(28,35,37,34)(32,40,36,39);(1,4)(2,6)(3,7)(5,8)(9,12,14,11)(10,17,15,16)(13,19,22,20)(18,23,21,24)(25,26)(27,31)(28,36)(29,30)(32,37)(33,38)(34,40)(35,39);(9,13,14,22)(10,18,15,21)(11,19,12,20)(16,23,17,24)(25,28,29,37)(26,32,30,36)(27,34,33,35)(31,39,38,40)

Polycyclic presentation:

2^9|b^2=gh,c^2=ghi,d^2=h,e^2=gh,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,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, 8607650);