#8662841

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

4571 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=h,e^2=g,f^2=h,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, 8570037);