#8873711

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

4504 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=i,c^2=i,d^2=i,e^2=i,f^2=g,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, 8780907);