#8659371

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

4479 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,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, 8566567);