#8289427

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

4566 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|b^2=h,c^2=gh,d^2=ghi,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,f^c=fgi,e^d=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8196623);