#9007674

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

4500 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=ghi,d^2=g,e^2=ghi,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,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, 8914870);