#8662919

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

4478 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)(3,8)(4,5)(6,7)(10,16)(11,12)(13,21)(22,23)(25,26,31,34)(27,30,32,28)(29,33,38,40)(35,39,37,36);(1,3,6,5)(2,4,7,8)(9,10,15,16)(11,14,12,19)(13,17,21,24)(18,23,20,22)(25,27)(26,30)(28,34)(29,37)(31,32)(33,36)(35,38)(39,40);(9,11,15,12)(10,14,16,19)(13,18,21,20)(17,23,24,22)(25,28,31,30)(26,32,34,27)(29,36,38,39)(33,37,40,35);(1,4,6,8)(2,3,7,5)(9,12,15,11)(10,14,16,19)(13,20,21,18)(17,23,24,22)(25,29)(26,33)(27,35)(28,36)(30,39)(31,38)(32,37)(34,40);(1,2)(3,4)(5,8)(6,7)(9,13)(10,17)(11,18)(12,20)(14,23)(15,21)(16,24)(19,22)(25,27)(26,28)(29,35)(30,34)(31,32)(33,36)(37,38)(39,40);(1,5,6,3)(2,4,7,8)(9,14)(10,12)(11,16)(13,22)(15,19)(17,18)(20,24)(21,23)(25,30,31,28)(26,32,34,27)(29,36,38,39)(33,35,40,37)

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=h,d^2=g,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,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, 8570115);