#8899305

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	#46368	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4315 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)(11,18)(12,17)(13,14)(15,16)(19,24)(20,21)(22,23)(25,26,31,29)(27,32,28,35)(30,37,38,33)(34,40,36,39);(1,2)(3,6)(4,5)(7,8)(10,16)(13,19)(17,18)(20,22)(25,27,31,28)(26,32,29,35)(30,34,38,36)(33,39,37,40);(2,5)(6,8)(9,11,15,12)(10,17,16,18)(13,20,19,22)(14,23,24,21)(25,28,31,27)(26,32,29,35)(30,36,38,34)(33,39,37,40);(3,7)(6,8)(9,12,15,11)(10,18,16,17)(13,20,19,22)(14,23,24,21);(1,3,4,7)(2,6,5,8)(9,13,15,19)(10,14,16,24)(11,20,12,22)(17,23,18,21)(25,29,31,26)(27,32,28,35)(30,37,38,33)(34,39,36,40);(1,2)(3,6)(4,5)(7,8)(9,14)(10,19)(11,21)(12,23)(13,16)(15,24)(17,20)(18,22)(25,30)(26,33)(27,34)(28,36)(29,37)(31,38)(32,39)(35,40)

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,c^2=h,d^2=g,e^2=hi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,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, 8806501);