#8158944

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

4280 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gi,c^2=hi,d^2=g,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi,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, 8066140);