#8188171

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

4457 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)(3,7)(4,6)(5,8)(9,10,15,14)(11,18,13,16)(12,20,22,17)(19,23,21,24)(25,26,32,36)(27,37,39,33)(28,34,29,31)(30,35,38,40);(1,2)(3,5)(4,6)(7,8)(9,11,15,13)(10,16,14,18)(12,19,22,21)(17,23,20,24)(25,27)(26,33)(28,35)(29,40)(30,34)(31,38)(32,39)(36,37);(1,2,4,6)(3,5,8,7)(10,14)(16,18)(17,20)(23,24)(25,28,32,29)(26,34,36,31)(27,35,39,40)(30,37,38,33);(9,12)(10,17)(11,19)(13,21)(14,20)(15,22)(16,23)(18,24)(25,29,32,28)(26,34,36,31)(27,35,39,40)(30,33,38,37);(1,3)(2,5)(4,8)(6,7)(9,13,15,11)(10,18,14,16)(12,21,22,19)(17,24,20,23)(25,30,32,38)(26,35,36,40)(27,34,39,31)(28,37,29,33);(1,2)(3,5)(4,6)(7,8)(9,14,15,10)(11,18,13,16)(12,20,22,17)(19,24,21,23)(25,31,32,34)(26,28,36,29)(27,38,39,30)(33,35,37,40)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8095367);