#8231393

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

4423 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:

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

Polycyclic presentation:

2^9|b^2=h,c^2=h,d^2=gh,e^2=g,f^2=i,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,f^c=fgi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8138589);