#8659373

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

4440 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=gh,c^2=g,d^2=g,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, 8566569);