#8607472

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

4491 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=hi,c^2=hi,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=egi,f^d=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8514668);