#9221184

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4546 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=ghi,c^2=gi,d^2=h,e^2=g,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, 9128380);