#8429404

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	C₂²×D₄	C₂⁴
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4179 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=g,c^2=hi,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, 8336600);