#417217

order	512
nilpotency class	3
isoclinism	isoclinic class of #397902
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#127
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	C₂×#399
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1031 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 324413);