#8891268

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

4314 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=h,c^2=ghi,d^2=g,e^2=ghi,f^2=hi,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, 8798464);