#9684528

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

4159 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)(12,13)(14,21)(15,17)(16,22)(18,23)(19,24)(25,26,32,31)(27,35,29,33)(28,34,39,38)(30,36,37,40);(1,2)(3,8)(4,6)(5,7)(9,11,15,22)(10,16,17,20)(12,21,23,19)(13,24,18,14)(25,27,32,29)(26,33,31,35)(28,37,39,30)(34,36,38,40);(1,3,5,6)(2,4,7,8)(10,17)(12,23)(16,20)(19,21)(25,28,32,39)(26,34,31,38)(27,37,29,30)(33,36,35,40);(9,12)(10,18)(11,21)(13,17)(14,16)(15,23)(19,22)(20,24)(25,29,32,27)(26,35,31,33)(28,30,39,37)(34,40,38,36);(1,2)(3,4)(5,7)(6,8)(9,13,15,18)(10,12,17,23)(11,14,22,24)(16,19,20,21)(25,30)(26,36)(27,28)(29,39)(31,40)(32,37)(33,38)(34,35);(1,4)(2,6)(3,7)(5,8)(9,14)(10,19)(11,13)(12,16)(15,24)(17,21)(18,22)(20,23)(25,31,32,26)(27,35,29,33)(28,38,39,34)(30,36,37,40)

Polycyclic presentation:

2^9|a^2=gh,b^2=g,c^2=ghi,d^2=gh,e^2=h,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,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, 9591724);