#409841

order	512
nilpotency class	3
isoclinism	isoclinic class of #393889
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂⁴	C₂²×D₄
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁵	C₂×D₄
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂²×#117 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1068 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=fg,b^2=h,c^2=fi,d^2=f,e^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,e^b=ei,d^c=di

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 317037);