#9545771

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

4235 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)(3,7)(4,6)(5,8)(9,10,13,16)(11,14,17,18)(12,15,20,22)(19,21,23,24)(25,26)(27,32)(28,33)(29,34)(30,31)(35,39)(36,40)(37,38);(1,2,4,6)(3,5,8,7)(9,10)(11,14)(12,15)(13,16)(17,18)(19,21)(20,22)(23,24)(25,27,31,28)(26,32,30,33)(29,36,37,35)(34,40,38,39);(1,2)(3,5)(4,6)(7,8)(11,17)(12,20)(14,18)(15,22)(26,30)(27,28)(29,37)(39,40);(1,3,4,8)(2,5,6,7)(9,10)(11,18)(12,15)(13,16)(14,17)(19,24)(20,22)(21,23)(25,28,31,27)(26,33,30,32)(29,35,37,36)(34,39,38,40);(9,11,13,17)(10,14,16,18)(12,19,20,23)(15,21,22,24)(25,29,31,37)(26,34,30,38)(27,35,28,36)(32,39,33,40);(9,12)(10,15)(11,19)(13,20)(14,21)(16,22)(17,23)(18,24)(25,30,31,26)(27,33,28,32)(29,38,37,34)(35,40,36,39)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9452967);