#8463722

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4386 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)(3,6)(4,7)(5,8)(10,11)(12,19)(16,18)(20,24)(25,26,32,30)(27,36,38,33)(28,39,34,29)(31,35,37,40);(3,8)(5,6)(9,10,14,11)(12,15,19,17)(13,16,21,18)(20,22,24,23)(25,27)(26,33)(28,37)(29,40)(30,36)(31,34)(32,38)(35,39);(1,3)(2,5)(4,8)(6,7)(25,28)(26,29)(27,37)(30,39)(31,38)(32,34)(33,40)(35,36);(1,3)(2,6)(4,8)(5,7)(9,11,14,10)(12,17,19,15)(13,16,21,18)(20,22,24,23)(25,29,32,39)(26,34,30,28)(27,35,38,40)(31,33,37,36);(9,12)(10,15)(11,17)(13,20)(14,19)(16,22)(18,23)(21,24)(25,30,32,26)(27,36,38,33)(28,39,34,29)(31,40,37,35);(1,2)(3,5)(4,7)(6,8)(9,13,14,21)(10,16,11,18)(12,20,19,24)(15,22,17,23)(25,31)(26,35)(27,34)(28,38)(29,36)(30,40)(32,37)(33,39)

Polycyclic presentation:

2^9|a^2=hi,b^2=ghi,d^2=g,e^2=hi,f^2=ghi,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,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, 8370918);