#8745960

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

4170 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=ghi,c^2=hi,d^2=hi,e^2=g,f^2=gh,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,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, 8653156);