#8383792

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

4454 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)(3,7)(4,6)(5,8)(9,10,15,16)(11,13,12,19)(14,17,22,23)(18,21,20,24)(25,26,31,35)(27,29,37,32)(28,34,30,33)(36,40,39,38);(1,2,4,6)(3,5,8,7)(10,16)(13,19)(17,23)(21,24)(25,27)(26,32)(28,36)(29,35)(30,39)(31,37)(33,40)(34,38);(1,2)(3,5)(4,6)(7,8)(9,11,15,12)(10,13,16,19)(14,20,22,18)(17,24,23,21)(25,28,31,30)(26,33,35,34)(27,36,37,39)(29,38,32,40);(1,3)(2,5)(4,8)(6,7)(9,12,15,11)(10,13,16,19)(14,20,22,18)(17,21,23,24)(25,29)(26,27)(28,38)(30,40)(31,32)(33,36)(34,39)(35,37);(9,13)(10,12)(11,16)(14,21)(15,19)(17,20)(18,23)(22,24)(25,30,31,28)(26,34,35,33)(27,36,37,39)(29,38,32,40);(9,14,15,22)(10,17,16,23)(11,18,12,20)(13,21,19,24)(25,27)(26,29)(28,39)(30,36)(31,37)(32,35)(33,40)(34,38)

Polycyclic presentation:

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

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, 8290988);