#8196365

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

4410 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8103561);