#8552461

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

4451 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)(11,16)(12,18)(13,14)(15,17)(19,21)(20,22)(23,24)(25,26,32,36)(27,35,30,31)(28,34,29,33)(37,38,39,40);(1,2)(3,5)(4,6)(7,8)(9,11,15,20)(10,16,17,22)(12,21,23,13)(14,18,19,24)(25,27,32,30)(26,31,36,35)(28,37,29,39)(33,38,34,40);(1,2,4,6)(3,5,7,8)(10,17)(11,20)(13,21)(18,24)(25,28,32,29)(26,33,36,34)(27,37,30,39)(31,38,35,40);(3,7)(5,8)(9,12)(10,18)(11,13)(14,16)(15,23)(17,24)(19,22)(20,21)(25,29,32,28)(26,34,36,33)(27,37,30,39)(31,38,35,40);(1,2)(3,5)(4,6)(7,8)(9,13)(10,19)(11,12)(14,17)(15,21)(16,24)(18,22)(20,23)(25,30,32,27)(26,35,36,31)(28,37,29,39)(33,38,34,40);(1,3,4,7)(2,5,6,8)(9,14,15,19)(10,13,17,21)(11,18,20,24)(12,22,23,16)(25,31,32,35)(26,27,36,30)(28,38,29,40)(33,37,34,39)

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=hi,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=dh,e^c=egi,f^d=fgi

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, 8459657);