#8722179

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

4442 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)(5,8)(9,10)(11,14)(12,17)(13,16)(15,20)(18,22)(19,24)(21,23)(25,26,30,31)(27,28,34,33)(29,32,39,40)(35,36,37,38);(1,2,4,6)(3,5,7,8)(10,11)(15,17)(16,18)(23,24)(25,26,30,31)(27,33,34,28)(29,32,39,40)(35,38,37,36);(1,3)(2,5)(4,7)(6,8)(10,11)(15,17)(16,18)(23,24)(26,31)(27,34)(32,40)(35,37);(1,2)(3,5)(4,6)(7,8)(9,11,14,10)(12,17,20,15)(13,16,22,18)(19,23,21,24)(25,27)(26,28)(29,37)(30,34)(31,33)(32,38)(35,39)(36,40);(1,3)(2,5)(4,7)(6,8)(9,12,14,20)(10,15,11,17)(13,21,22,19)(16,24,18,23)(25,28)(26,27)(29,38)(30,33)(31,34)(32,37)(35,40)(36,39);(9,13,14,22)(10,16,11,18)(12,19,20,21)(15,23,17,24)(25,29,30,39)(26,32,31,40)(27,35,34,37)(28,36,33,38)

Polycyclic presentation:

2^9|a^2=i,b^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,d^c=dgi,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, 8629375);