#8689818

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

4632 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=g,d^2=h,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,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, 8597014);