#9713715

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	#48765	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4230 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=hi,d^2=h,e^2=gi,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, 9620911);