#8429429

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	#2537	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4303 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=gh,c^2=hi,d^2=h,e^2=h,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,f^d=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, 8336625);