#411122

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

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂⁴	C₂²×D₄
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂×#376 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

902 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=ghi,c^2=fgi,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, 318318);