#9683533

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	#46353	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4344 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:

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

Polycyclic presentation:

2^9|b^2=gh,c^2=gi,d^2=h,e^2=g,f^2=h,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, 9590729);