#9712865

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

4591 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=h,d^2=h,e^2=g,f^2=ghi,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, 9620061);