#8254562

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

4428 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:

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

Polycyclic presentation:

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

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, 8161758);