#8340202

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

4464 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=g,e^2=gi,f^2=gi,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, 8247398);