#9221217

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	#38020	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4684 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:

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

Polycyclic presentation:

2^9|b^2=gi,c^2=hi,d^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,e^d=egi,f^d=fgi,f^e=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9128413);