#8634234

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

4321 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=h,c^2=gh,d^2=g,e^2=g,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=egi,f^d=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, 8541430);