#8398626

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

4493 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)(11,17,12,16)(13,21,22,18)(19,23,20,24)(25,26)(27,32)(28,29)(30,33)(31,34)(35,39)(36,37)(38,40);(1,2)(3,4)(5,6)(7,8)(9,11,15,12)(10,16,14,17)(13,20,22,19)(18,24,21,23)(25,26,31,34)(27,32,28,29)(30,33,38,40)(35,39,36,37);(2,6)(4,7)(10,14)(16,17)(18,21)(23,24)(25,27,31,28)(26,32,34,29)(30,36,38,35)(33,37,40,39);(2,6)(3,8)(9,12,15,11)(10,17,14,16)(13,19,22,20)(18,23,21,24)(25,28,31,27)(26,32,34,29)(30,36,38,35)(33,39,40,37);(1,3)(2,7)(4,6)(5,8)(9,13,15,22)(10,18,14,21)(11,19,12,20)(16,23,17,24)(25,29,31,32)(26,28,34,27)(30,37,38,39)(33,36,40,35);(1,4,5,7)(2,3,6,8)(9,14,15,10)(11,17,12,16)(13,21,22,18)(19,24,20,23)(25,30,31,38)(26,33,34,40)(27,35,28,36)(29,37,32,39)

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=gh,d^2=g,e^2=g,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,e^b=egi,f^c=fgi,e^d=egi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8305822);