#8288377

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
prime radical	#43119	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4312 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=g,d^2=ghi,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,f^c=fgi,e^d=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8195573);