#8942967

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

4661 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=gi,d^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,e^c=egi,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, 8850163);