#8760690

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

4386 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=h,c^2=h,d^2=gh,e^2=g,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^c=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, 8667886);