#8429322

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

4163 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=hi,d^2=g,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, 8336518);