#9713745

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

4340 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)(3,7)(4,5)(6,8)(9,10)(11,21)(12,18)(13,19)(14,24)(15,17)(16,22)(20,23)(26,28)(27,30)(33,36)(35,37);(3,8)(6,7)(9,11,15,22)(10,16,17,21)(12,20,19,24)(13,23,18,14)(25,26,31,28)(27,34,30,32)(29,36,38,33)(35,39,37,40);(1,3)(2,6)(4,7)(5,8)(10,17)(12,19)(16,21)(20,24)(25,27,31,30)(26,32,28,34)(29,35,38,37)(33,39,36,40);(1,2,5,4)(3,7,8,6)(9,12,15,19)(10,18,17,13)(11,20,22,24)(14,21,23,16)(25,28,31,26)(27,32,30,34)(29,36,38,33)(35,39,37,40);(1,3)(2,7)(4,6)(5,8)(9,13,15,18)(10,19,17,12)(11,14,22,23)(16,20,21,24)(25,29)(26,33)(27,35)(28,36)(30,37)(31,38)(32,39)(34,40);(1,4,5,2)(3,7,8,6)(9,14)(10,20)(11,13)(12,21)(15,23)(16,19)(17,24)(18,22)(25,30,31,27)(26,32,28,34)(29,37,38,35)(33,39,36,40)

Polycyclic presentation:

2^9|b^2=g,c^2=ghi,d^2=h,e^2=hi,f^2=i,b^a=bg,c^a=ch,d^a=di,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, 9620941);