#9164007

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	#44103	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4321 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=ghi,c^2=h,d^2=g,e^2=ghi,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^d=egi,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, 9071203);