#8254305

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

4451 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)(11,20)(12,16)(13,22)(14,18)(15,19)(17,23)(21,24)(26,32)(27,29)(28,36)(35,40);(1,2,5,4)(3,7,8,6)(11,12)(13,23)(16,20)(17,22)(25,26,31,32)(27,33,29,39)(28,37,36,30)(34,35,38,40);(2,4)(6,7)(9,11,15,12)(10,16,19,20)(13,21,23,18)(14,17,24,22)(25,27,31,29)(26,33,32,39)(28,34,36,38)(30,40,37,35);(9,12,15,11)(10,16,19,20)(13,18,23,21)(14,17,24,22)(25,28)(26,30)(27,34)(29,38)(31,36)(32,37)(33,40)(35,39);(1,3,5,8)(2,6,4,7)(9,13,15,23)(10,17,19,22)(11,21,12,18)(14,16,24,20)(25,29,31,27)(26,33,32,39)(28,38,36,34)(30,40,37,35);(1,4,5,2)(3,7,8,6)(9,14,15,24)(10,18,19,21)(11,22,12,17)(13,16,23,20)(25,30)(26,28)(27,35)(29,40)(31,37)(32,36)(33,38)(34,39)

Polycyclic presentation:

2^9|b^2=hi,c^2=h,d^2=gh,e^2=gi,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, 8161501);