#9065683

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

4577 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)(11,20)(12,22)(13,19)(14,16)(15,17)(18,23)(21,24)(25,26,31,35)(27,33,28,32)(29,30,34,40)(36,39,38,37);(1,2)(3,4)(5,6)(7,8)(9,11,15,14)(10,16,17,20)(12,21,23,19)(13,18,24,22)(25,27,31,28)(26,32,35,33)(29,36,34,38)(30,37,40,39);(2,6)(3,7)(10,17)(13,24)(16,20)(18,22)(25,27,31,28)(26,33,35,32)(29,38,34,36)(30,37,40,39);(3,7)(4,8)(9,12)(10,18)(11,21)(13,20)(14,19)(15,23)(16,24)(17,22)(25,28,31,27)(26,33,35,32)(29,36,34,38)(30,37,40,39);(1,3)(2,4)(5,7)(6,8)(9,13,15,24)(10,19,17,21)(11,22,14,18)(12,16,23,20)(25,29,31,34)(26,30,35,40)(27,36,28,38)(32,37,33,39);(1,4)(2,3)(5,8)(6,7)(9,14,15,11)(10,20,17,16)(12,21,23,19)(13,18,24,22)(25,30)(26,34)(27,37)(28,39)(29,35)(31,40)(32,38)(33,36)

Polycyclic presentation:

2^9|a^2=gh,b^2=g,c^2=gh,d^2=gh,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,e^c=egi,e^d=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, 8972879);