#9706960

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

4168 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:

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

Polycyclic presentation:

2^9|b^2=gh,c^2=gi,d^2=h,e^2=ghi,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, 9614156);