#8875529

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

4289 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8782725);