#8899335

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

4358 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:

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

Polycyclic presentation:

2^9|a^2=g,c^2=g,d^2=gh,e^2=ghi,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,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, 8806531);