#9230866

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

4383 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)(3,4)(5,6)(7,8)(9,10,16,14)(11,18,21,15)(12,17,13,22)(19,24,20,23)(27,33)(30,34)(35,39)(37,40);(2,6)(4,7)(9,11,16,21)(10,15,14,18)(12,19,13,20)(17,23,22,24)(25,26,31,28)(27,30,33,34)(29,36,38,32)(35,40,39,37);(1,2,5,6)(3,7,8,4)(9,12,16,13)(10,17,14,22)(11,19,21,20)(15,23,18,24)(25,27)(26,30)(28,34)(29,35)(31,33)(32,37)(36,40)(38,39);(1,3)(2,7)(4,6)(5,8)(9,13,16,12)(10,17,14,22)(11,20,21,19)(15,23,18,24)(25,28,31,26)(27,34,33,30)(29,36,38,32)(35,40,39,37);(1,4,5,7)(2,3,6,8)(9,14,16,10)(11,18,21,15)(12,22,13,17)(19,24,20,23)(25,29,31,38)(26,32,28,36)(27,35,33,39)(30,37,34,40);(1,3)(2,7)(4,6)(5,8)(9,15,16,18)(10,11,14,21)(12,23,13,24)(17,19,22,20)(25,30,31,34)(26,33,28,27)(29,37,38,40)(32,39,36,35)

Polycyclic presentation:

2^9|a^2=gi,b^2=gh,c^2=g,d^2=gh,e^2=ghi,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,e^d=egi,f^d=fgi,f^e=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, 9138062);