#8289324

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	C₂×#121	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4193 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=g,c^2=h,d^2=hi,e^2=h,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, 8196520);