#9716179

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

4166 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=g,d^2=g,e^2=hi,f^2=ghi,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,d^c=dh,f^e=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, 9623375);