#9224387

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

4494 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)(3,4)(5,6)(7,8)(9,10,15,13)(11,19,22,16)(12,21,23,17)(14,18,24,20)(25,26,31,32)(27,34,28,33)(29,30,39,40)(35,38,37,36);(2,6)(4,7)(9,11,15,22)(10,16,13,19)(12,20,23,18)(14,21,24,17)(26,32)(29,39)(33,34)(35,37);(1,2,5,6)(3,7,8,4)(25,27,31,28)(26,33,32,34)(29,35,39,37)(30,36,40,38);(1,3)(2,7)(4,6)(5,8)(9,12,15,23)(10,17,13,21)(11,20,22,18)(14,19,24,16)(25,28,31,27)(26,34,32,33)(29,35,39,37)(30,36,40,38);(1,4,5,7)(2,3,6,8)(9,13,15,10)(11,19,22,16)(12,17,23,21)(14,18,24,20)(25,29,31,39)(26,30,32,40)(27,35,28,37)(33,36,34,38);(1,3)(2,7)(4,6)(5,8)(9,14)(10,18)(11,21)(12,16)(13,20)(15,24)(17,22)(19,23)(25,30)(26,29)(27,36)(28,38)(31,40)(32,39)(33,35)(34,37)

Polycyclic presentation:

2^9|a^2=gi,b^2=gh,c^2=h,d^2=gi,e^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, 9131583);