#8873798

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

4361 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=hi,c^2=ghi,d^2=gi,e^2=i,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^d=egi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8780994);