#8898278

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

4468 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)(13,18,22,21)(19,24,20,23)(25,26,32,36)(27,37,40,33)(28,31,30,34)(29,39,38,35);(1,2)(3,5)(4,6)(7,8)(9,11,15,12)(10,16,14,17)(13,19,22,20)(18,23,21,24)(25,27,32,40)(26,33,36,37)(28,38,30,29)(31,39,34,35);(1,2,4,6)(3,5,7,8)(9,11,15,12)(10,17,14,16)(13,19,22,20)(18,24,21,23)(25,28,32,30)(26,34,36,31)(27,38,40,29)(33,35,37,39);(3,7)(5,8)(9,12,15,11)(10,17,14,16)(13,19,22,20)(18,23,21,24)(25,29)(26,35)(27,28)(30,40)(31,33)(32,38)(34,37)(36,39);(1,3)(2,5)(4,7)(6,8)(9,13)(10,18)(11,19)(12,20)(14,21)(15,22)(16,23)(17,24)(25,30,32,28)(26,34,36,31)(27,38,40,29)(33,39,37,35);(1,2)(3,5)(4,6)(7,8)(9,14,15,10)(11,17,12,16)(13,21,22,18)(19,24,20,23)(25,31,32,34)(26,28,36,30)(27,39,40,35)(29,33,38,37)

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=gi,d^2=gh,e^2=h,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, 8805474);