#9064635

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

4635 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=gh,e^2=gh,f^2=g,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, 8971831);