#8898291

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

4690 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

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

Polycyclic presentation:

2^9|a^2=gh,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, 8805487);