#9224256

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

4631 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=g,c^2=g,d^2=gi,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, 9131452);