#8963645

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

4507 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=ghi,c^2=h,d^2=gi,e^2=gi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,e^c=egi,f^c=fgi,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, 8870841);