#9707106

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
prime radical	#2535	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4130 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:

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

Polycyclic presentation:

2^9|a^2=h,c^2=gi,d^2=g,e^2=i,f^2=gh,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, 9614302);