#203145

order	512
nilpotency class	3
isoclinism	isoclinic class of #19304
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#121
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₂×#388
prime radical	#3444	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

694 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=eg,c^2=e,d^2=h,e^2=h,f^2=i,g^2=hi,b^a=be,c^a=cf,d^a=di,e^a=eh,f^a=fi,d^b=dh,d^c=dh,f^c=fhi

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, 110341);