#9685521

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

4459 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)(3,8)(4,6)(5,7)(9,10)(11,19)(12,17)(13,21)(14,15)(16,20)(18,23)(22,24)(25,26,31,34)(27,29,33,39)(28,30,38,32)(35,40,36,37);(1,3,5,4)(2,6,7,8)(10,15)(12,22)(16,19)(18,21)(25,27,31,33)(26,29,34,39)(28,36,38,35)(30,37,32,40);(10,15)(11,20)(17,24)(18,21)(25,28,31,38)(26,32,34,30)(27,35,33,36)(29,37,39,40);(1,3,5,4)(2,6,7,8)(9,11,14,20)(10,16,15,19)(12,21,22,18)(13,17,23,24)(25,28)(26,30)(27,36)(29,37)(31,38)(32,34)(33,35)(39,40);(1,2)(3,6)(4,8)(5,7)(9,12,14,22)(10,17,15,24)(11,21,20,18)(13,16,23,19)(25,29,31,39)(26,33,34,27)(28,37,38,40)(30,35,32,36);(1,4,5,3)(2,6,7,8)(9,13,14,23)(10,18,15,21)(11,17,20,24)(12,19,22,16)(25,30)(26,28)(27,37)(29,36)(31,32)(33,40)(34,38)(35,39)

Polycyclic presentation:

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

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, 9592717);